Integration by Substitution, Parts & Partial Fractions

This section contains documents created from scanned original files and other
documents that could not be made accessible to screen reader software. A "#"
symbol is used to denote such documents.

Lecture Notes

Antidifferentiation: Integration by Substitution

PDF
Section 4, Page 3 to page 4

Step-by-step guide for integrating using the substitution method. Examples include finding the antiderivative of x*sin(x2) and the antiderivative of sin(x)3*cos(x).

Instructor: Prof. Jason Starr
Prior Knowledge: Antidifferentiation (sections 2 and 3 of this lecture)

Change of Variables

PDF
Section 5, Page 4 to page 5

Using substitution of variables to evaluate definite integrals, including change of limits. Includes worked example.

Instructor: Prof. Jason Starr
Prior Knowledge: Riemann Integral (section 4 of lecture 14) and Fundamental Theorem of Calculus (section 3 of lecture 15)

Review of Inverse Substitution and Another Example

PDF
Section 1, Page 1 to page 4

Step-by-step method of inverse substitution with example.

Instructor: Prof. Jason Starr
Prior Knowledge: Inverse Substitution (section 3 of lecture 25)

Antidifferentiating Simple Rational Expressions

PDF
Section 2, Page 4 to page 5

Definition of rational expressions and partial fractions. Formulas for integrating partial fractions.

Instructor: Prof. Jason Starr
Prior Knowledge: Inverse Substitution (section 1 of this lecture)

Simplifying Rational Expressions: Division and Factoring

PDF
Section 3, Page 5 to page 6

Method of using polynomial division and factoring to simplify a rational expression. Includes example of reducing (x3 + 1) / (x2 + 3x + 2).

Instructor: Prof. Jason Starr
Prior Knowledge: Simple Rational Expressions (section 2 of this lecture)

Simplifying Rational Expressions: Partial Fraction Decomposition

PDF
Section 4, Page 6 to page 7

Method of partial fraction decomposition, with example 1 / (1-x2).

Instructor: Prof. Jason Starr
Prior Knowledge: Simplifying Rational Expressions (section 3 of this lecture)

The Heaviside Cover-up Method

PDF
Section 5, Page 7 to page 9

The cover-up method for finding the coefficients in a partial fraction decomposition, with example z2 / (1 - z2)2.

Instructor: Prof. Jason Starr
Prior Knowledge: Partial Fraction Decomposition (section 4 of this lecture)

Integration by Parts

PDF
Section 1, Page 1 to page 2

Introduction to method of integration by parts, with example of integrating x*cos(x).

Instructor: Prof. Jason Starr
Prior Knowledge: Product Rule (section 4 of lecture 3) and Differentials (section 1 of lecture 13)

How to Use Integration by Parts

PDF
Section 2, Page 2 to page 3

Further explanation of integration by parts, with example of integrating ln(x).

Instructor: Prof. Jason Starr
Prior Knowledge: Integration by Parts (section 1 of this lecture)

Reduction Formulas

PDF
Section 2, Page 3 to page 4

Definition of reduction formulas found using integration by parts, with examples of reduction formulas for integrating (ln(x))n and (tn)*(et).

Instructor: Prof. Jason Starr
Prior Knowledge: Integration by Parts (sections 1 and 2 of this lecture)

PDF
Section 3, Page 4 to page 5

Derivation of reduction formula for integrating (sin(x))n.

Instructor: Prof. Jason Starr
Prior Knowledge: Reduction Formulas (section 2 of this lecture)

Review Problems

PDF
Section 3, Page 3 to page 4

Problems and answers without full explanation. Finding tangent lines to an ellipse, minimizing surface area of a grain silo, finding the volume of a solid of revolution, computing an antiderivative using trig substitution, and computing an antiderivative using integration by parts.

Instructor: Prof. Jason Starr
Prior Knowledge: Tangent Lines (section 1 of lecture 2), Max/Min Problems (section 2 of lecture 10), Volume of Solids of Revolution (section 3 of lecture 19), Inverse Substitution (section 3 of lecture 25), Integration by Parts (section 1 of lecture 27)

Heaviside's Cover-Up Method

PDF
Section , Page 1 to page 3

Definition and explanation of this method for partial fractions, including four examples.

Instructor: Prof. David Jerison
Prior Knowledge: None

Online Textbook Chapters

Substitution

Document

Anti-differentiation by applying the chain rule backwards, including a list of classes of functions that can be integrated using this method of substitution.

Instructor: Prof. Daniel J. Kleitman
Prior Knowledge: Anti-derivatives (OT19.1)

Integration by Parts

Document

Anti-differentiation using the backward version of the product rule, including an example.

Instructor: Prof. Daniel J. Kleitman
Prior Knowledge: Anti-derivatives (OT19.1)

Partial Fraction Decomposition

Document

Finding anti-derivatives of rational functions using the method of partial fractions.

Instructor: Prof. Daniel J. Kleitman
Prior Knowledge: Anti-derivatives (OT19.1)

Basic Techniques for Integrals

Document

Rules for integrating polynomials and other simple integrals by inspection, as well as techniques for integrating by substitution, parts, and partial fractions.

Instructor: Prof. Daniel J. Kleitman
Prior Knowledge: Anti-derivatives (OT19.1)

Practice Problem

Partial Fractions and the Substitution Method

PDF
Problem 3 (page 2)

Two part question which involves a basic example of partial fractions and an application of the substitution method for integration.

Instructor: Prof. Jason Starr
Prior Knowledge: None

Exam Questions

Evaluating Definite Integrals

PDF
Problem 4 (page 6 to page 7)

Five-part problem evaluating integrals involving the substitution method, logarithmic functions, and trigonometric functions.

Instructor: Prof. Jason Starr
Prior Knowledge: None

Miscellaneous Integration Problems

PDF
Problem I.1 (page 1) to problem IV.5 (page 4)

Eighteen problems with answers but not complete solutions on these four topics.

Instructor: Prof. Jason Starr
Prior Knowledge: None

Integration by Parts

PDF
Problem 1 (page 2)

Computing an antiderivative using the method of integration by parts.

Instructor: Prof. Jason Starr
Prior Knowledge: None

Partial Fractions

PDF
Problem 3 (page 4)

Finding the partial fraction decomposition of a fraction of two polynomials and using it to find the antiderivative of that function.

Instructor: Prof. Jason Starr
Prior Knowledge: None

Techniques of Antidifferentiation

PDF
Problem 4 (page 5)

Evaluating an antiderivative that requires the application of multiple techniques.

Instructor: Prof. Jason Starr
Prior Knowledge: None

Integration by Parts

PDF
Problem 1 (page 1) to problem 2 (page 1)

Evaluating a definite and indefinite integral using the method of integration by parts.

Instructor: Prof. Jason Starr
Prior Knowledge: None

Trigonometric Substitution

PDF
Problem 7 (page 1)

Evaluating an integral using the method of trigonometric substitution.

Instructor: Prof. Jason Starr
Prior Knowledge: None

Techniques of Antidifferentiation

PDF
Problem 12 (page 2)

Evaluating four integrals using multiple techniques.

Instructor: Prof. Jason Starr
Prior Knowledge: None

Integration by Substitution

PDF
Problem 4.4 (page 3) to problem 4.5 (page 3)

Two problems which involve evaluating a definite integral.

Instructor: Prof. Jason Starr
Prior Knowledge: None

Antiderivatives of Inverse Trigonometric Functions

PDF
Problem 6.4 (page 5) to problem 6.7 (page 5)

Four questions which involve evaluating antiderivatives of the inverse sine, cosine, and tangent functions.

Instructor: Prof. Jason Starr
Prior Knowledge: None

Definite Integrals

PDF
Problem 1 (page 1)

Two integrals to be evaluated.

Instructor: Prof. David Jerison
Prior Knowledge: None

Definite Integrals

PDF
Problem 1 (page 1)

Two integrals to be evaluated.

Prior Knowledge: None

Indefinite Integrals: Ratio of Polynomials

PDF
Problem 1 (page 1)

Antidifferentiating a function which is a ratio of polynomials.

Instructor: Prof. David Jerison
Prior Knowledge: None

Trigonometric Substitution

PDF
Problem 2 (page 1)

Evaluating a definite integral using a suggested trigonometric substitution.

Instructor: Prof. David Jerison
Prior Knowledge: None

Reduction Formulas

PDF
Problem 3 (page 1)

Finding a reduction formula for two integrals involving exponentials.

Instructor: Prof. David Jerison
Prior Knowledge: None

Trigonometric Substitution

PDF
Problem 1 (page 1)

Evaluating a definite integral using a trigonometric substitution.

Instructor: Prof. David Jerison
Prior Knowledge: None

Indefinite Integrals: Ratio of Polynomials

PDF
Problem 3 (page 1)

Antidifferentiating a function which is a ratio of polynomials.

Instructor: Prof. David Jerison
Prior Knowledge: None

Indefinite Integrals

PDF
Problem 1 (page 1) to problem 2 (page 1)

Two questions which involve evaluating indefinite integrals using advanced techniques.

Instructor: Prof. David Jerison
Prior Knowledge: None

Trigonometric Substitution

PDF
Problem 3 (page 1)

Evaluating a definite integral using a trigonometric substitution.

Instructor: Prof. David Jerison
Prior Knowledge: None

Evaluating Integrals

PDF
Problem 12 (page 2)

Two integrals to be evaluated, one involving a ratio of polynomials, the other involving a natural logarithm.

Instructor: Prof. David Jerison
Prior Knowledge: None

Trigonometric Substitution

PDF
Problem 13 (page 2)

Evaluating a definite integral using the trigonometric substitution of the tangent function.

Instructor: Prof. David Jerison
Prior Knowledge: None

Differentials and Indefinite Integration

PDF - 2.2 MB
Problem 3A-1 (page 21) to problem 3A-3 (page 21)

Three questions which involve evaluating five differentials and twenty indefinite integrals using a range of techniques.

Instructor: Prof. David Jerison
Prior Knowledge: None

Change of Variables and Estimating Integrals

PDF - 2.2 MB
Problem 3E-1 (page 24) to problem 3E-7 (page 25)

Seven questions which involve evaluating or estimating integrals by using the method of substitution of variables.

Instructor: Prof. David Jerison
Prior Knowledge: None

Integration by Direct Substitution

PDF - 2.2 MB
Problem 5B-1 (page 36) to problem 5B-16 (page 36)

Sixteen integrals to be evaluated using the method of substitution.

Instructor: Prof. David Jerison
Prior Knowledge: None

Trigonometric Integrals

PDF - 2.2 MB
Problem 5C-1 (page 36) to problem 5C-14 (page 36)

Fourteen integrals to be evaluated, each of which involves a trigonometric function.

Instructor: Prof. David Jerison
Prior Knowledge: None

Integration by Inverse Substitution

PDF - 2.2 MB
Problem 5D-1 (page 36) to problem 5D-15 (page 37)

Fifteen integrals to be evaluated using the method of inverse substitution and completing the square.

Instructor: Prof. David Jerison
Prior Knowledge: None

Integration by Partial Fractions

PDF - 2.2 MB
Problem 5E-1 (page 37) to problem 5E-13 (page 38)

Thirteen questions which involve integrals that must be evaluated using the method of partial fractions.

Instructor: Prof. David Jerison
Prior Knowledge: None