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8.5: Integration by Parts


We have already seen that recognizing the product rule can be useful, when we noticed that

[int sec^3u+sec u an^2u,du=sec u an u.]

As with substitution, we do not have to rely on insight or cleverness to discover such antiderivatives; there is a technique that will often help to uncover the product rule.

Start with the product rule:

[{dover dx}f(x)g(x)=f'(x)g(x)+f(x)g'(x).]

We can rewrite this as

[f(x)g(x)=int f'(x)g(x),dx +int f(x)g'(x),dx,]

and then

[int f(x)g'(x),dx=f(x)g(x)-int f'(x)g(x),dx.]

This may not seem particularly useful at first glance, but it turns out that in many cases we have an integral of the form

[int f(x)g'(x),dx]

but that

[int f'(x)g(x),dx]

is easier. This technique for turning one integral into another is called integration by parts, and is usually written in more compact form. If we let (u=f(x)) and (v=g(x)) then (du=f'(x),dx) and (dv=g'(x),dx) and

[int u,dv = uv-int v,du.]

To use this technique we need to identify likely candidates for (u=f(x)) and (dv=g'(x),dx).

Example (PageIndex{1})

Evaluate (displaystyle int xln x,dx).

Solution

Let (u=ln x) so (du=1/x,dx). Then we must let (dv=x,dx) so ( v=x^2/2) and

[ int xln x,dx={x^2ln xover 2}-int {x^2over2}{1over x},dx= {x^2ln xover 2}-int {xover2},dx={x^2ln xover 2}-{x^2over4}+C. ]

Example (PageIndex{2})

Evaluate (displaystyle int xsin x,dx).

Solution

Let (u=x) so (du=dx). Then we must let (dv=sin x,dx) so (v=-cos x) and

[int xsin x,dx=-xcos x-int -cos x,dx= -xcos x+int cos x,dx=-xcos x+sin x+C.]

Example (PageIndex{3})

Evaluate (displaystyle intsec^3 x,dx).

Solution

Of course we already know the answer to this, but we needed to be clever to discover it. Here we'll use the new technique to discover the antiderivative. Let (u=sec x) and ( dv=sec^2 x,dx). Then (du=sec x an x,dx) and (v= an x) and

[eqalign{ intsec^3 x,dx&=sec x an x-int an^2xsec x,dxcr &=sec x an x-int (sec^2x-1)sec x,dxcr &=sec x an x-int sec^3x,dx +intsec x,dx.cr }]

At first this looks useless---we're right back to ( intsec^3x,dx). But looking more closely:

[eqalign{ intsec^3x,dx&=sec x an x-int sec^3x,dx +intsec x,dxcr intsec^3x,dx+int sec^3x,dx&=sec x an x +intsec x,dxcr 2intsec^3x,dx&=sec x an x +intsec x,dxcr intsec^3x,dx&={sec x an xover2} +{1over2}intsec x,dxcr &={sec x an xover2} +{ln|sec x+ an x|over2}+C.cr }]

Example (PageIndex{4})

Evaluate ( displaystyle int x^2sin x,dx).

Solution

Let (u=x^2), (dv=sin x,dx); then (du=2x,dx) and (v=-cos x). Now ( int x^2sin x,dx=-x^2cos x+int 2xcos x,dx). This is better than the original integral, but we need to do integration by parts again. Let (u=2x), (dv=cos x,dx); then (du=2) and (v=sin x), and

[eqalign{ int x^2sin x,dx&=-x^2cos x+int 2xcos x,dxcr &=-x^2cos x+ 2xsin x - int 2sin x,dxcr &=-x^2cos x+ 2xsin x + 2cos x + C.cr }]

Such repeated use of integration by parts is fairly common, but it can be a bit tedious to accomplish, and it is easy to make errors, especially sign errors involving the subtraction in the formula. There is a nice tabular method to accomplish the calculation that minimizes the chance for error and speeds up the whole process. We illustrate with the previous example. Here is the table:

sign(u)

(dv)

(x^2)(sin x)
(-)(2x)(-cos x)
(2)(-sin x)
(-)(0)(cos x)
or
(u)(dv)
(x^2)(sin x)
(-2x)(-cos x)
(2)(-sin x)
(0)(cos x)

To form the first table, we start with (u) at the top of the second column and repeatedly compute the derivative; starting with (dv) at the top of the third column, we repeatedly compute the antiderivative. In the first column, we place a "(-)'' in every second row. To form the second table we combine the first and second columns by ignoring the boundary; if you do this by hand, you may simply start with two columns and add a "(-)'' to every second row.

To compute with this second table we begin at the top. Multiply the first entry in column (u) by the second entry in column (dv) to get ( -x^2cos x), and add this to the integral of the product of the second entry in column (u) and second entry in column (dv). This gives:

[-x^2cos x+int 2xcos x,dx,]

or exactly the result of the first application of integration by parts. Since this integral is not yet easy, we return to the table. Now we multiply twice on the diagonal, ( (x^2)(-cos x)) and ((-2x)(-sin x)) and then once straight across, ((2)(-sin x)), and combine these as

[-x^2cos x+2xsin x-int 2sin x,dx,]

giving the same result as the second application of integration by parts. While this integral is easy, we may return yet once more to the table. Now multiply three times on the diagonal to get ( (x^2)(-cos x)), ((-2x)(-sin x)), and ((2)(cos x)), and once straight across, ((0)(cos x)). We combine these as before to get

[ -x^2cos x+2xsin x +2cos x+int 0,dx= -x^2cos x+2xsin x +2cos x+C. ]

Typically we would fill in the table one line at a time, until the "straight across'' multiplication gives an easy integral. If we can see that the (u) column will eventually become zero, we can instead fill in the whole table; computing the products as indicated will then give the entire integral, including the "(+C,)'', as above.


Integration by parts

Figure 5.4 shows a technique called integration by parts. If the integral is easier than the integral , then we can calculate the easier one, and then by simple geometry determine the one we wanted. Identifying the large rectangle that surrounds both shaded areas, and the small white rectangle on the lower left, we have

In the case of an indefinite integral, we have a similar relationship de- rived from the product rule:

Integrating both sides, we have the following relation.


Integration by parts

Since a definite integral can always be done by evaluating an indefinite integral at its upper and lower limits, one usually uses this form. Integrals don't usually come prepackaged in a form that makes it obvious that you should use integration by parts. What the equation for integration by parts tells us is that if we can split up the integrand into two factors, one of which (the ) we know how to integrate, we have the option of changing the integral into a new form in which that factor becomes its integral, and the other factor becomes its derivative. If we choose the right way of splitting up the integrand into parts, the result can be a simplification.


How do you evaluate the integral of #(ln x)^2 dx#?

bp has one great solution Method 1. There are other solutions:

Both of the solution presented below use Integration by Parts.
I use the form:

Both of the solution presented below use #int lnx dx = xlnx - x +C# , which can be done by integration by parts. (And, of course, verified by differentiating the answer.)

Then #du = (2lnx)/x dx# and #v = x#

Integration by parts gives us:

#int (lnx)^2 dx = x(lnx)^2 - 2int lnx dx# #

#color(white)"sssssss"# # =x(lnx)^2-2(xlnx - x) +C#

#color(white)"sssssss"# # =x(lnx)^2-2xlnx + 2x +C#

#int (lnx)^2 dx = int (lnx)(lnx)dx#

So, #du = 1/x dx# and #v= xlnx -x#

The parts formula gives us:

#int (lnx)^2 dx = (lnx)(xlnx -x)-int(xlnx-x)/x dx#

#color(white)"sssssss"# # =x(lnx)^2-xlnx -int (color(red)(lnx) - color(green)(1))dx#

#color(white)"sssssss"# # =x(lnx)^2-xlnx -(color(red)(xlnx-x) - color(green)(x)) +C#


AP Calculus BC - A. Tennyson

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Intro to Course/Calendars/Unit Reviews

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Calculus BC

Unit 1 Limits and Continuity

Class Notes (you may choose the video or mp4 version of each topic)

  • You belong to 2nd Period
  • It is before September 22 2020, 6:15 PM

Unit 2 Derivatives

Class Notes (you may choose the video or mp4 version of each topic)

Unit 3 Chain Rule, Implicit, Related Rates

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Unit 4 Derivatives of Transcendental Functions

Unit 5 Applications of Derivatives

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Unit 6 Area and Integrals

Unit 7 u-substitution

Unit 8 Differential Equations

Unit 9 Area and Volume

Unit 10 -Integration by Parts, L'Hopital's Rule, Improper Integrals

Unit 11- Series Tests

Unit 12- Taylor Polynomials

Unit 13- Vectors and Polar

AP Test Reviews

Mrs. Tennyson


Amie Tennyson
[email protected]
817-698-5659

Calculus BC Tutorials:
Tuesdays & Thursdays 4-4:30


AP Calculus BC - A. Tennyson

Students , click here for a list of all of the BNHS teacher's advisory Zoom links! Advisory meets every Monday and Friday 9:30-10!

Open all Close all

Instructions: Clicking on the section name will show / hide the section.

Intro to Course/Calendars/Unit Reviews

Northwest ISD is not responsible for content on external websites or services.

Calculus BC

Unit 1 Limits and Continuity

Class Notes (you may choose the video or mp4 version of each topic)

  • You belong to 2nd Period
  • It is before September 22 2020, 6:15 PM

Unit 2 Derivatives

Class Notes (you may choose the video or mp4 version of each topic)

Unit 3 Chain Rule, Implicit, Related Rates

Class Notes (you may choose the video or mp4 version of each topic)

Unit 4 Derivatives of Transcendental Functions

Unit 5 Applications of Derivatives

Class Notes (you may choose the video or mp4 version of each topic)

Unit 6 Area and Integrals

Unit 7 u-substitution

Unit 8 Differential Equations

Unit 9 Area and Volume

Unit 10 -Integration by Parts, L'Hopital's Rule, Improper Integrals

Unit 11- Series Tests

Unit 12- Taylor Polynomials

Unit 13- Vectors and Polar

AP Test Reviews

Mrs. Tennyson


Amie Tennyson
[email protected]
817-698-5659

Calculus BC Tutorials:
Tuesdays & Thursdays 4-4:30


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    • Antiderivatives, integration, and the fundamental theorem of calculus
    • Indeterminate forms and L'Hopital's rule
    • Sequences and series
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    • Parametric equations and polar coordinates
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    Table of Contents

    1. The Basics

    • 1.1 Overview
      • 1.1.1 An Introduction to Thinkwell Calculus
      • 1.1.2 The Two Questions of Calculus
      • 1.1.3 Average Rates of Change
      • 1.1.4 How to Do Math
      • 1.2 Precalculus Review
        • 1.2.1 Functions
        • 1.2.2 Graphing Lines
        • 1.2.3 Parabolas
        • 1.2.4 Some Non-Euclidean Geometry

        2. Limits

        • 2.1 The Concept of the Limit
          • 2.1.1 Finding Rate of Change over an Interval
          • 2.1.2 Finding Limits Graphically
          • 2.1.3 The Formal Definition of a Limit
          • 2.1.4 The Limit Laws, Part I
          • 2.1.5 The Limit Laws, Part II
          • 2.1.6 One-Sided Limits
          • 2.1.7 The Squeeze Theorem
          • 2.1.8 Continuity and Discontinuity
          • 2.2 Evaluating Limits
            • 2.2.1 Evaluating Limits
            • 2.2.2 Limits and Indeterminate Forms
            • 2.2.3 Two Techniques for Evaluating Limits
            • 2.2.4 An Overview of Limits

            3. An Introduction to Derivatives

            • 3.1 Understanding the Derivative
              • 3.1.1 Rates of Change, Secants, and Tangents
              • 3.1.2 Finding Instantaneous Velocity
              • 3.1.3 The Derivative
              • 3.1.4 Differentiability
              • 3.2 Using the Derivative
                • 3.2.1 The Slope of a Tangent Line
                • 3.2.2 Instantaneous Rate
                • 3.2.3 The Equation of a Tangent Line
                • 3.2.4 More on Instantaneous Rate
                • 3.3 Some Special Derivatives
                  • 3.3.1 The Derivative of the Reciprocal Function
                  • 3.3.2 The Derivative of the Square Root Function

                  4. Computational Techniques

                  • 4.1 The Power Rule
                    • 4.1.1 A Shortcut for Finding Derivatives
                    • 4.1.2 A Quick Proof of the Power Rule
                    • 4.1.3 Uses of the Power Rule
                    • 4.2 The Product and Quotient Rules
                      • 4.2.1 The Product Rule
                      • 4.2.2 The Quotient Rule
                      • 4.3 The Chain Rule
                        • 4.3.1 An Introduction to the Chain Rule
                        • 4.3.2 Using the Chain Rule
                        • 4.3.3 Combining Computational Techniques

                        5. Special Functions

                        • 5.1 Trigonometric Functions
                          • 5.1.1 A Review of Trigonometry
                          • 5.1.2 Graphing Trigonometric Functions
                          • 5.1.3 The Derivatives of Trigonometric Functions
                          • 5.1.4 The Number Pi
                          • 5.2 Exponential Functions
                            • 5.2.1 Graphing Exponential Functions
                            • 5.2.2 Derivatives of Exponential Functions
                            • 5.2.3 The Music of Math
                            • 5.3 Logarithmic Functions
                              • 5.3.1 Evaluating Logarithmic Functions
                              • 5.3.2 The Derivative of the Natural Log Function
                              • 5.3.3 Using the Derivative Rules with Transcendental Functions

                              6. Implicit Differentiation

                              • 6.1 Implicit Differentiation Basics
                                • 6.1.1 An Introduction to Implicit Differentiation
                                • 6.1.2 Finding the Derivative Implicitly
                                • 6.2 Applying Implicit Differentiation
                                  • 6.2.1 Using Implicit Differentiation
                                  • 6.2.2 Applying Implicit Differentiation

                                  7. Applications of Differentiation

                                  • 7.1 Position and Velocity
                                    • 7.1.1 Acceleration and the Derivative
                                    • 7.1.2 Solving Word Problems Involving Distance and Velocity
                                    • 7.2 Linear Approximation
                                      • 7.2.1 Higher-Order Derivatives and Linear Approximation
                                      • 7.2.2 Using the Tangent Line Approximation Formula
                                      • 7.2.3 Newton's Method
                                      • 7.3 Related Rates
                                        • 7.3.1 The Pebble Problem
                                        • 7.3.2 The Ladder Problem
                                        • 7.3.3 The Baseball Problem
                                        • 7.3.4 The Blimp Problem
                                        • 7.3.5 Math Anxiety
                                        • 7.4 Optimization
                                          • 7.4.1 The Connection Between Slope and Optimization
                                          • 7.4.2 The Fence Problem
                                          • 7.4.3 The Box Problem
                                          • 7.4.4 The Can Problem
                                          • 7.4.5 The Wire-Cutting Problem

                                          8. Curve Sketching

                                          • 8.1 Introduction
                                            • 8.1.1 An Introduction to Curve Sketching
                                            • 8.1.2 Three Big Theorems
                                            • 8.1.3 Morale Moment
                                            • 8.2 Critical Points
                                              • 8.2.1 Critical Points
                                              • 8.2.2 Maximum and Minimum
                                              • 8.2.3 Regions Where a Function Increases or Decreases
                                              • 8.2.4 The First Derivative Test
                                              • 8.2.5 Math Magic
                                              • 8.3 Concavity
                                                • 8.3.1 Concavity and Inflection Points
                                                • 8.3.2 Using the Second Derivative to Examine Concavity
                                                • 8.3.3 The Möbius Band
                                                • 8.4 Graphing Using the Derivative
                                                  • 8.4.1 Graphs of Polynomial Functions
                                                  • 8.4.2 Cusp Points and the Derivative
                                                  • 8.4.3 Domain-Restricted Functions and the Derivative
                                                  • 8.4.4 The Second Derivative Test
                                                  • 8.5 Asymptotes
                                                    • 8.5.1 Vertical Asymptotes
                                                    • 8.5.2 Horizontal Asymptotes and Infinite Limits
                                                    • 8.5.3 Graphing Functions with Asymptotes
                                                    • 8.5.4 Functions with Asymptotes and Holes
                                                    • 8.5.5 Functions with Asymptotes and Critical Points

                                                    9. The Basics of Integration

                                                    • 9.1 Antiderivatives
                                                      • 9.1.1 Antidifferentiation
                                                      • 9.1.2 Antiderivatives of Powers of x
                                                      • 9.1.3 Antiderivatives of Trigonometric and Exponential Functions
                                                      • 9.2 Integration by Substitution
                                                        • 9.2.1 Undoing the Chain Rule
                                                        • 9.2.2 Integrating Polynomials by Substitution
                                                        • 9.3 Illustrating Integration by Substitution
                                                          • 9.3.1 Integrating Composite Trigonometric Functions by Substitution
                                                          • 9.3.2 Integrating Composite Exponential and Rational Functions by Substitution
                                                          • 9.3.3 More Integrating Trigonometric Functions by Substitution
                                                          • 9.3.4 Choosing Effective Function Decompositions
                                                          • 9.4 The Fundamental Theorem of Calculus
                                                            • 9.4.1 Approximating Areas of Plane Regions
                                                            • 9.4.2 Areas, Riemann Sums, and Definite Integrals
                                                            • 9.4.3 The Fundamental Theorem of Calculus, Part I
                                                            • 9.4.4 The Fundamental Theorem of Calculus, Part II
                                                            • 9.4.5 Illustrating the Fundamental Theorem of Calculus
                                                            • 9.4.6 Evaluating Definite Integrals

                                                            10. Applications of Integration

                                                            • 10.1 Motion
                                                              • 10.1.1 Antiderivatives and Motion
                                                              • 10.1.2 Gravity and Vertical Motion
                                                              • 10.1.3 Solving Vertical Motion Problems
                                                              • 10.2 Finding the Area between Two Curves
                                                                • 10.2.1 The Area between Two Curves
                                                                • 10.2.2 Limits of Integration and Area
                                                                • 10.2.3 Common Mistakes to Avoid When Finding Areas
                                                                • 10.2.4 Regions Bound by Several Curves
                                                                • 10.3 Integrating with Respect to y
                                                                  • 10.3.1 Finding Areas by Integrating with Respect to y: Part One
                                                                  • 10.3.2 Finding Areas by Integrating with Respect to y: Part Two
                                                                  • 10.3.3 Area, Integration by Substitution, and Trigonometry

                                                                  11. Calculus I Review

                                                                  12. Math Fun

                                                                  • 12.1 Paradoxes
                                                                    • 12.1.1 An Introduction to Paradoxes
                                                                    • 12.1.2 Paradoxes and Air Safety
                                                                    • 12.1.3 Newcomb's Paradox
                                                                    • 12.1.4 Zeno's Paradox
                                                                    • 12.2 Sequences
                                                                      • 12.2.1 Fibonacci Numbers
                                                                      • 12.2.2 The Golden Ratio

                                                                      13. An Introduction to Calculus II

                                                                      • 13.1 Introduction
                                                                        • 13.1.1 Welcome to Calculus II
                                                                        • 13.1.2 Review: Calculus I in 20 Minutes

                                                                        14. L'Hôpital's Rule

                                                                        • 14.1 Indeterminate Quotients
                                                                          • 14.1.1 Indeterminate Forms
                                                                          • 14.1.2 An Introduction to L'Hôpital's Rule
                                                                          • 14.1.3 Basic Uses of L'Hôpital's Rule
                                                                          • 14.1.4 More Exotic Examples of Indeterminate Forms
                                                                          • 14.2 Other Indeterminate Forms
                                                                            • 14.2.1 L'Hôpital's Rule and Indeterminate Products
                                                                            • 14.2.2 L'Hôpital's Rule and Indeterminate Differences
                                                                            • 14.2.3 L'Hôpital's Rule and One to the Infinite Power
                                                                            • 14.2.4 Another Example of One to the Infinite Power

                                                                            15. Elementary Functions and Their Inverses

                                                                            • 15.1 Inverse Functions
                                                                              • 15.1.1 The Exponential and Natural Log Functions
                                                                              • 15.1.2 Differentiating Logarithmic Functions
                                                                              • 15.1.3 Logarithmic Differentiation
                                                                              • 15.1.4 The Basics of Inverse Functions
                                                                              • 15.1.5 Finding the Inverse of a Function
                                                                              • 15.2 The Calculus of Inverse Functions
                                                                                • 15.2.1 Derivatives of Inverse Functions
                                                                                • 15.3 Inverse Trigonometric Functions
                                                                                  • 15.3.1 The Inverse Sine, Cosine, and Tangent Functions
                                                                                  • 15.3.2 The Inverse Secant, Cosecant, and Cotangent Functions
                                                                                  • 15.3.3 Evaluating Inverse Trigonometric Functions
                                                                                  • 15.4 The Calculus of Inverse Trigonometric Functions
                                                                                    • 15.4.1 Derivatives of Inverse Trigonometric Functions
                                                                                    • 15.4.2 More Calculus of Inverse Trigonometric Functions
                                                                                    • 15.5 The Hyperbolic Functions
                                                                                      • 15.5.1 Defining the Hyperbolic Functions
                                                                                      • 15.5.2 Hyperbolic Identities
                                                                                      • 15.5.3 Derivatives of Hyperbolic Functions

                                                                                      16. Techniques of Integration

                                                                                      • 16.1 Integration Using Tables
                                                                                        • 16.1.1 An Introduction to the Integral Table
                                                                                        • 16.1.2 Making u-Substitutions
                                                                                        • 16.2 Integrals Involving Powers of Sine and Cosine
                                                                                          • 16.2.1 An Introduction to Integrals with Powers of Sine and Cosine
                                                                                          • 16.2.2 Integrals with Powers of Sine and Cosine
                                                                                          • 16.2.3 Integrals with Even and Odd Powers of Sine and Cosine
                                                                                          • 16.3 Integrals Involving Powers of Other Trigonometric Functions
                                                                                            • 16.3.1 Integrals of Other Trigonometric Functions
                                                                                            • 16.3.2 Integrals with Odd Powers of Tangent and Any Power of Secant
                                                                                            • 16.3.3 Integrals with Even Powers of Secant and Any Power of Tangent
                                                                                            • 16.4 An Introduction to Integration by Partial Fractions
                                                                                              • 16.4.1 Finding Partial Fraction Decompositions
                                                                                              • 16.4.2 Partial Fractions
                                                                                              • 16.4.3 Long Division
                                                                                              • 16.5 Integration by Partial Fractions with Repeated Factors
                                                                                                • 16.5.1 Repeated Linear Factors: Part One
                                                                                                • 16.5.2 Repeated Linear Factors: Part Two
                                                                                                • 16.5.3 Distinct and Repeated Quadratic Factors
                                                                                                • 16.5.4 Partial Fractions of Transcendental Functions
                                                                                                • 16.6 Integration by Parts
                                                                                                  • 16.6.1 An Introduction to Integration by Parts
                                                                                                  • 16.6.2 Applying Integration by Parts to the Natural Log Function
                                                                                                  • 16.6.3 Inspirational Examples of Integration by Parts
                                                                                                  • 16.6.4 Repeated Application of Integration by Parts
                                                                                                  • 16.6.5 Algebraic Manipulation and Integration by Parts
                                                                                                  • 16.7 An Introduction to Trigonometric Substitution
                                                                                                    • 16.7.1 Converting Radicals into Trigonometric Expressions
                                                                                                    • 16.7.2 Using Trigonometric Substitution to Integrate Radicals
                                                                                                    • 16.7.3 Trigonometric Substitutions on Rational Powers
                                                                                                    • 16.8 Trigonometric Substitution Strategy
                                                                                                      • 16.8.1 An Overview of Trigonometric Substitution Strategy
                                                                                                      • 16.8.2 Trigonometric Substitution Involving a Definite Integral: Part One
                                                                                                      • 16.8.3 Trigonometric Substitution Involving a Definite Integral: Part Two
                                                                                                      • 16.9 Numerical Integration
                                                                                                        • 16.9.1 Deriving the Trapezoidal Rule
                                                                                                        • 16.9.2 An Example of the Trapezoidal Rule

                                                                                                        17. Improper Integrals

                                                                                                        • 17.1 Improper Integrals
                                                                                                          • 17.1.1 The First Type of Improper Integral
                                                                                                          • 17.1.2 The Second Type of Improper Integral
                                                                                                          • 17.1.3 Infinite Limits of Integration, Convergence, and Divergence

                                                                                                          18. Applications of Integral Calculus

                                                                                                          • 18.1 The Average Value of a Function
                                                                                                            • 18.1.1 Finding the Average Value of a Function
                                                                                                            • 18.2 Finding Volumes Using Cross-Sections
                                                                                                              • 18.2.1 Finding Volumes Using Cross-Sectional Slices
                                                                                                              • 18.2.2 An Example of Finding Cross-Sectional Volumes
                                                                                                              • 18.3 Disks and Washers
                                                                                                                • 18.3.1 Solids of Revolution
                                                                                                                • 18.3.2 The Disk Method along the y-Axis
                                                                                                                • 18.3.3 A Transcendental Example of the Disk Method
                                                                                                                • 18.3.4 The Washer Method across the x-Axis
                                                                                                                • 18.3.5 The Washer Method across the y-Axis
                                                                                                                • 18.4 Shells
                                                                                                                  • 18.4.1 Introducing the Shell Method
                                                                                                                  • 18.4.2 Why Shells Can Be Better Than Washers
                                                                                                                  • 18.4.3 The Shell Method: Integrating with Respect to y
                                                                                                                  • 18.5 Arc Lengths and Functions
                                                                                                                    • 18.5.1 An Introduction to Arc Length
                                                                                                                    • 18.5.2 Finding Arc Lengths of Curves Given by Functions
                                                                                                                    • 18.6 Work
                                                                                                                      • 18.6.1 An Introduction to Work
                                                                                                                      • 18.6.2 Calculating Work
                                                                                                                      • 18.6.3 Hooke's Law
                                                                                                                      • 18.7 Moments and Centers of Mass
                                                                                                                        • 18.7.1 Center of Mass
                                                                                                                        • 18.7.2 The Center of Mass of a Thin Plate

                                                                                                                        19. Sequences and Series

                                                                                                                        • 19.1 Sequences
                                                                                                                          • 19.1.1 The Limit of a Sequence
                                                                                                                          • 19.1.2 Determining the Limit of a Sequence
                                                                                                                          • 19.1.3 The Squeeze and Absolute Value Theorems
                                                                                                                          • 19.2 Monotonic and Bounded Sequences
                                                                                                                            • 19.2.1 Monotonic and Bounded Sequences
                                                                                                                            • 19.3 Infinite Series
                                                                                                                              • 19.3.1 An Introduction to Infinite Series
                                                                                                                              • 19.3.2 The Summation of Infinite Series
                                                                                                                              • 19.3.3 Geometric Series
                                                                                                                              • 19.3.4 Telescoping Series
                                                                                                                              • 19.4 Convergence and Divergence
                                                                                                                                • 19.4.1 Properties of Convergent Series
                                                                                                                                • 19.4.2 The nth-Term Test for Divergence
                                                                                                                                • 19.5 The Integral Test
                                                                                                                                  • 19.5.1 An Introduction to the Integral Test
                                                                                                                                  • 19.5.2 Examples of the Integral Test
                                                                                                                                  • 19.5.3 Using the Integral Test
                                                                                                                                  • 19.5.4 Defining p-Series
                                                                                                                                  • 19.6 The Direct Comparison Test
                                                                                                                                    • 19.6.1 An Introduction to the Direct Comparison Test
                                                                                                                                    • 19.6.2 Using the Direct Comparison Test
                                                                                                                                    • 19.7 The Limit Comparison Test
                                                                                                                                      • 19.7.1 An Introduction to the Limit Comparison Test
                                                                                                                                      • 19.7.2 Using the Limit Comparison Test
                                                                                                                                      • 19.7.3 Inverting the Series in the Limit Comparison Test
                                                                                                                                      • 19.8 The Alternating Series
                                                                                                                                        • 19.8.1 Alternating Series
                                                                                                                                        • 19.8.2 The Alternating Series Test
                                                                                                                                        • 19.8.3 Estimating the Sum of an Alternating Series
                                                                                                                                        • 19.9 Absolute and Conditional Convergences
                                                                                                                                          • 19.9.1 Absolute and Conditional Convergence
                                                                                                                                          • 19.10 The Ratio and Root Tests
                                                                                                                                            • 19.10.1 The Ratio Test
                                                                                                                                            • 19.10.2 Examples of the Ratio Test
                                                                                                                                            • 19.10.3 The Root Test
                                                                                                                                            • 19.11 Polynomial Approximations of Elementary Functions
                                                                                                                                              • 19.11.1 Polynomial Approximation of Elementary Functions
                                                                                                                                              • 19.11.2 Higher-Degree Approximations
                                                                                                                                              • 19.12 Taylor and Maclaurin Polynomials
                                                                                                                                                • 19.12.1 Taylor Polynomials
                                                                                                                                                • 19.12.2 Maclaurin Polynomials
                                                                                                                                                • 19.12.3 The Remainder of a Taylor Polynomial
                                                                                                                                                • 19.12.4 Approximating the Value of a Function
                                                                                                                                                • 19.13 Taylor and Maclaurin Series
                                                                                                                                                  • 19.13.1 Taylor Series
                                                                                                                                                  • 19.13.2 Examples of the Taylor and Maclaurin Series
                                                                                                                                                  • 19.13.3 New Taylor Series
                                                                                                                                                  • 19.13.4 The Convergence of Taylor Series
                                                                                                                                                  • 19.14 Power Series
                                                                                                                                                    • 19.14.1 The Definition of Power Series
                                                                                                                                                    • 19.14.2 The Interval and Radius of Convergence
                                                                                                                                                    • 19.14.3 Finding the Interval and Radius of Convergence: Part One
                                                                                                                                                    • 19.14.4 Finding the Interval and Radius of Convergence: Part Two
                                                                                                                                                    • 19.14.5 Finding the Interval and Radius of Convergence: Part Three
                                                                                                                                                    • 19.15 Power Series Representations of Functions
                                                                                                                                                      • 19.15.1 Differentiation and Integration of Power Series
                                                                                                                                                      • 19.15.2 Finding Power Series Representations by Differentiation
                                                                                                                                                      • 19.15.3 Finding Power Series Representations by Integration
                                                                                                                                                      • 19.15.4 Integrating Functions Using Power Series

                                                                                                                                                      20. Differential Equations

                                                                                                                                                      • 20.1 Separable Differential Equations
                                                                                                                                                        • 20.1.1 An Introduction to Differential Equations
                                                                                                                                                        • 20.1.2 Solving Separable Differential Equations
                                                                                                                                                        • 20.1.3 Finding a Particular Solution
                                                                                                                                                        • 20.1.4 Direction Fields
                                                                                                                                                        • 20.2 Solving a Homogeneous Differential Equation
                                                                                                                                                          • 20.2.1 Separating Homogeneous Differential Equations
                                                                                                                                                          • 20.2.2 Change of Variables
                                                                                                                                                          • 20.3 Growth and Decay Problems
                                                                                                                                                            • 20.3.1 Exponential Growth
                                                                                                                                                            • 20.3.2 Radioactive Decay
                                                                                                                                                            • 20.4 Solving First-Order Linear Differential Equations
                                                                                                                                                              • 20.4.1 First-Order Linear Differential Equations
                                                                                                                                                              • 20.4.2 Using Integrating Factors

                                                                                                                                                              21. Parametric Equations and Polar Coordinates

                                                                                                                                                              • 21.1 Understanding Parametric Equations
                                                                                                                                                                • 21.1.1 An Introduction to Parametric Equations
                                                                                                                                                                • 21.1.2 The Cycloid
                                                                                                                                                                • 21.1.3 Eliminating Parameters
                                                                                                                                                                • 21.2 Calculus and Parametric Equations
                                                                                                                                                                  • 21.2.1 Derivatives of Parametric Equations
                                                                                                                                                                  • 21.2.2 Graphing the Elliptic Curve
                                                                                                                                                                  • 21.2.3 The Arc Length of a Parameterized Curve
                                                                                                                                                                  • 21.2.4 Finding Arc Lengths of Curves Given by Parametric Equations
                                                                                                                                                                  • 21.3 Understanding Polar Coordinates
                                                                                                                                                                    • 21.3.1 The Polar Coordinate System
                                                                                                                                                                    • 21.3.2 Converting between Polar and Cartesian Forms
                                                                                                                                                                    • 21.3.3 Spirals and Circles
                                                                                                                                                                    • 21.3.4 Graphing Some Special Polar Functions
                                                                                                                                                                    • 21.4 Polar Functions and Slope
                                                                                                                                                                      • 21.4.1 Calculus and the Rose Curve
                                                                                                                                                                      • 21.4.2 Finding the Slopes of Tangent Lines in Polar Form
                                                                                                                                                                      • 21.5 Polar Functions and Area
                                                                                                                                                                        • 21.5.1 Heading toward the Area of a Polar Region
                                                                                                                                                                        • 21.5.2 Finding the Area of a Polar Region: Part One
                                                                                                                                                                        • 21.5.3 Finding the Area of a Polar Region: Part Two
                                                                                                                                                                        • 21.5.4 The Area of a Region Bounded by Two Polar Curves: Part One
                                                                                                                                                                        • 21.5.5 The Area of a Region Bounded by Two Polar Curves: Part Two

                                                                                                                                                                        22. Vector Calculus and the Geometry of R 2 and R 3

                                                                                                                                                                        • 22.1 Vectors and the Geometry of R 2 and R 3
                                                                                                                                                                          • 22.1.1 Coordinate Geometry in Three-Dimensional Space
                                                                                                                                                                          • 22.1.2 Introduction to Vectors
                                                                                                                                                                          • 22.1.3 Vectors in R 2 and R 3
                                                                                                                                                                          • 22.1.4 An Introduction to the Dot Product
                                                                                                                                                                          • 22.1.5 Orthogonal Projections
                                                                                                                                                                          • 22.1.6 An Introduction to the Cross Product
                                                                                                                                                                          • 22.1.7 Geometry of the Cross Product
                                                                                                                                                                          • 22.1.8 Equations of Lines and Planes in R 3
                                                                                                                                                                          • 22.2 Vector Functions
                                                                                                                                                                            • 22.2.1 Introduction to Vector Functions
                                                                                                                                                                            • 22.2.2 Derivatives of Vector Functions
                                                                                                                                                                            • 22.2.3 Vector Functions: Smooth Curves
                                                                                                                                                                            • 22.2.4 Vector Functions: Velocity and Acceleration

                                                                                                                                                                            About the Author

                                                                                                                                                                            Edward Burger is an award-winning professor with a passion for teaching mathematics.

                                                                                                                                                                            Since 2013, Edward Burger has been President of Southwestern University, a top-ranked liberal arts college in Georgetown, Texas. Previously, he was Professor of Mathematics at Williams College. Dr. Burger earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

                                                                                                                                                                            Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.


                                                                                                                                                                            Keywords

                                                                                                                                                                            This work was supported by the Natural Sciences and Engineering Research Council (NSERC) , the Canada Research Chairs program , Bombardier Aerospace , Mathematics of Information Technology and Complex Systems (MITACS) , and the University of Toronto .

                                                                                                                                                                            Rensselaer Polytechnic Institute, Department of Mechanical, Aerospace, and Nuclear Engineering.

                                                                                                                                                                            Tier 1 Canada Research Chair in Computational Aerodynamics, J. Armand Bombardier Foundation Chair in Aerospace Flight.


                                                                                                                                                                            Math 231/249, Honors Calculus II

                                                                                                                                                                            A second course in calculus, focusing on sequences and series, but also covering techniques of integration, parametric equations, polar coordinates, and complex numbers. While covering the same basic material as the standard sections, this honors class does so in more detail, including some additional topics. As such, it is for those students who, regardless of their major, are particularly interested in, and excited by, mathematics. In addition, a score of 5 on the AP Calculus AB exam, or a grade of "A" in Math 220 or 221 is required for enrollment. Students who enroll in this course must also register for Math 249 Q1H, the "honors supplement", with CRN 32044.

                                                                                                                                                                            Grading:

                                                                                                                                                                            • Weekly Homework (15%). Homework will be assigned during each lecture and due at the beginning of class each Wednesday. No late homework will be accepted however, your lowest homework grade will be dropped so you are effectively allowed one infinitely late assignment. Collaboration on homework is permitted, nay encouraged. However, you must write up your solutions individually and understand them completely. You may use a computer or calculator on the HW for experimentation and to check your answers, but may not refer to it directly in the solution, e.g. "by Mathematica" is not an acceptable justification for deriving one equation from another. (Also, computers and calculators will not be allowed on the exams, so it's best not to get too dependent on them.)
                                                                                                                                                                            • Three in-class exams (20% each). These will be closed-book, calculator-free exams, though you will be allowed to bring one piece of paper with handwritten formulas. They will be on Fridays, in particular, September 19, October 17, and November 14.
                                                                                                                                                                            • A final exam (25%) Our final exam is scheduled for Friday, December 12 from 1:30-4:30 pm.

                                                                                                                                                                            Textbook

                                                                                                                                                                            • Smith and Minton, Calculus: Early Transcendental Functions, 3rd edition, McGraw Hill, 2006 or 2007.

                                                                                                                                                                            We will be covering Chapters 6-9, so either the "Single Variable" or "Full" version of this book is fine. As to the future value of the longer version for those planning on taking Math 241 (Calculus III), the honors sections of that course do not use this text, though some, but not all, of the standard sections do.


                                                                                                                                                                            8.5: Integration by Parts

                                                                                                                                                                            Consider a discrete grid consisting of points and uniform spacing on some, possibly unbounded, domain .

                                                                                                                                                                            Definition 17. A difference operator approximating is said to satisfy SBP on the domain with respect to a positive definite scalar product ,

                                                                                                                                                                            This is the discrete counterpart of integration by parts for the operator,

                                                                                                                                                                            If the interval is infinite, say or , certain fall-off conditions are required and Eq. (8.22 ) replaced by dropping the corresponding boundary term(s).

                                                                                                                                                                            Example 50. Standard centered differences as defined by Eq. (8.19 ) in the domain or for periodic domains and functions satisfy SBP with respect to the trivial scalar product (),

                                                                                                                                                                            The scalar product or associated norm are said to be diagonal if

                                                                                                                                                                            Accuracy and Efficiency.

                                                                                                                                                                            When constructing SBP operators, the discrete scalar product cannot be arbitrarily fixed and afterward the difference operator solved for so that it satisfies the SBP property (8.22 ) &ndash in general this leads to no solutions. The coefficients of and those of have to be simultaneously solved for. The resulting systems of equations lead to SBP operators being in general not unique, with increasing freedom with the accuracy order. In the diagonal case the resulting norm is automatically positive definite but not so in the full-restricted case.

                                                                                                                                                                            We label the operators by their order of accuracy in the interior and near boundary points. For diagonal norms and restricted full ones this would be and , respectively.

                                                                                                                                                                            Example 51. : For the simplest case, , the SBP operator and scalar product are unique:

                                                                                                                                                                            The operator and its associated scalar product are also unique in the diagonal norm case:

                                                                                                                                                                            On the other hand, the operators have one, three and ten free parameters, respectively. Up to their associated scalar products are unique, while for one of the free parameters enters in . For the full-restricted case, have three, four and five free parameters, respectively, all of which appear in the corresponding scalar products.

                                                                                                                                                                            A possibility [396 ] is to use the non-uniqueness of SBP operators to minimize the boundary stencil s ize . If the difference operator in the interior is a standard centered difference with accuracy-order then there are points at and near each boundary, where the accuracy is of order (with in the diagonal case and in the full restricted one). The integer can be referred to as the b oundary width. The boundary stencil size is the number of gridpoints that the difference operator uses to evaluate its approximation at those boundary points.

                                                                                                                                                                            However, minimizing such size, as well as any naive or arbitrary choice of the free parameters, easily leads to a large spectral radius and as a consequence restrictive CFL (see Section 7 ) limit in the case of explicit evolutions. Sometimes it also leads to rather large boundary truncation errors. Thus, an alternative is to numerically compute the spectral radius for these multi-parameter families of SBP operators and find in each case the parameter choice that leads to a minimum [399 , 281 ] . It turns out that in this way the order of accuracy can be increased from the very low one of to higher-order ones such as or with a very small change in the CFL limit. It involves some work, but since the SBP property (8.22 ) is independent of the system of equations one wants to solve, it only needs to be done once. In the full-restricted case, when marching through parameter space and minimizing the spectral radius, this minimization has to be constrained with the condition that the resulting norm is actually positive definite.

                                                                                                                                                                            The non-uniqueness of high-order SBP operators can be further used to minimize a combination of the average of the boundary truncation error (ABTE), defined below, without a significant increase in the spectral radius. For definiteness consider a left boundary. If a Taylor expansion of the FD operator is written as


                                                                                                                                                                            Watch the video: How To Integrate Using U-Substitution (December 2021).