- Antidifferentiation: Integration by Substitution
- Change of Variables
- Review of Inverse Substitution and Another Example
- Antidifferentiating Simple Rational Expressions
- Simplifying Rational Expressions: Division and Factoring
- Simplifying Rational Expressions: Partial Fraction Decomposition
- The Heaviside Cover-up Method
- Integration by Parts
- How to Use Integration by Parts
- Reduction Formulas
- Advanced Reduction Formulas
- Review Problems
- Substitution
- Partial Fraction Decomposition
- Basic Techniques for Integrals
- Partial Fractions and the Substitution Method
- Evaluating Definite Integrals

- Miscellaneous Integration Problems
- Partial Fractions
- Techniques of Antidifferentiation
- Trigonometric Substitution
- Integration by Substitution
- Antiderivatives of Inverse Trigonometric Functions
- Definite Integrals
- Indefinite Integrals: Ratio of Polynomials
- Indefinite Integrals
- Evaluating Integrals
- Differentials and Indefinite Integration
- Change of Variables and Estimating Integrals
- Integration by Direct Substitution
- Trigonometric Integrals
- Integration by Inverse Substitution
- Integration by Partial Fractions
- Integration by Parts and Reduction Formulas

## Antidifferentiation: Integration by Substitution

Step-by-step guide for integrating using the substitution method. Examples include finding the antiderivative of x*sin(x^{2}) and the antiderivative of sin(x)^{3}*cos(x).

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

## Change of Variables

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

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

## Review of Inverse Substitution and Another Example

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

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

## Antidifferentiating Simple Rational Expressions

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

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

## Simplifying Rational Expressions: Division and Factoring

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

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

## Simplifying Rational Expressions: Partial Fraction Decomposition

Method of partial fraction decomposition, with example 1 / (1-x^{2}).

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

## The Heaviside Cover-up Method

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

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

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

18.01

*Single Variable Calculus*, Fall 2006

Prof. David Jerison

**Course Material Related to This Topic:**

## Integration by Parts

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

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

Computing an antiderivative using the method of integration by parts.

- Complete exam problem 1 on page 2
- Check solution to exam problem 1 on page 1

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

- Complete exam problems 1 to 2 on page 1
- Check solution to exam problems 1 to 2 on page 1

18.013A

*Calculus with Applications*, Spring 2005

Prof. Daniel J. Kleitman

**Course Material Related to This Topic:**

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

## How to Use Integration by Parts

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

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

## Reduction Formulas

Definition of reduction formulas found using integration by parts, with examples of reduction formulas for integrating (ln(x))^{n} and (t^{n})*(e^{t}).

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

Finding a reduction formula for two integrals involving exponentials.

18.01

*Single Variable Calculus*, Fall 2006

Prof. David Jerison

**Course Material Related to This Topic:**

- Complete exam problem 3 on page 1
- Check solution to exam problem 3 on page 1

## Advanced Reduction Formulas

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

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

## Review Problems

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.

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

## Substitution

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

18.013A

*Calculus with Applications*, Spring 2005

Prof. Daniel J. Kleitman

**Course Material Related to This Topic:**

## Partial Fraction Decomposition

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

18.013A

*Calculus with Applications*, Spring 2005

Prof. Daniel J. Kleitman

**Course Material Related to This Topic:**

## Basic Techniques for Integrals

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

18.013A

*Calculus with Applications*, Spring 2005

Prof. Daniel J. Kleitman

**Course Material Related to This Topic:**

## Partial Fractions and the Substitution Method

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

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

- Complete practice problem 3 on page 2
- Check solution to practice problem 3 on pages 3–4

## Evaluating Definite Integrals

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

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

- Complete exam problem 4 on pages 6–7
- Check solution to exam problem 4 on pages 3–6

## Miscellaneous Integration Problems

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

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

- Complete exam problem I.1 on page 1 to problem IV.5 on page 4
- Check solution to exam problems on pages 1–4

## Partial Fractions

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

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

- Complete exam problem 3 on page 4
- Check solution to exam problem 3 on pages 2–3

## Techniques of Antidifferentiation

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

Evaluating an antiderivative that requires the application of multiple techniques.

- Complete exam problem 4 on page 5
- Check solution to exam problem 4 on pages 3–4

Evaluating four integrals using multiple techniques.

- Complete exam problem 12 on page 2
- Check solution to exam problem 12 on pages 4–5

## Trigonometric Substitution

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

Evaluating an integral using the method of trigonometric substitution.

- Complete exam problem 7 on page 1
- Check solution to exam problem 7 on page 3

18.01

*Single Variable Calculus*, Fall 2006

Prof. David Jerison

**Course Material Related to This Topic:**

Evaluating a definite integral using a suggested trigonometric substitution.

- Complete exam problem 2 on page 1
- Check solution to exam problem 2 on page 1
- Complete exam problem 1 on page 1
- Check solution to exam problem 1 on page 1
- Complete exam problem 3 on page 1
- Check solution to exam problem 3 on page 1

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

- Complete exam problem 13 on page 2
- Check solution to exam problem 13 on page 1

## Integration by Substitution

Two problems which involve evaluating a definite integral.

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

- Complete exam problems 4.4 to 4.5 on page 3
- Check solution to exam problems 4.4 to 4.5 on page 3

## Antiderivatives of Inverse Trigonometric Functions

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

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

- Complete exam problems 6.4 to 6.7 on page 5
- Check solution to exam problems 6.4 to 6.7 on page 5

## Definite Integrals

Two integrals to be evaluated.

18.01

*Single Variable Calculus*, Fall 2006

Prof. David Jerison

**Course Material Related to This Topic:**

- Complete exam problem 1 on page 1
- Check solution to exam problem 1 on page 1
- Complete exam problem 1 on page 1
- Check solution to exam problem 1 on page 1

## Indefinite Integrals: Ratio of Polynomials

Antidifferentiating a function which is a ratio of polynomials.

18.01

*Single Variable Calculus*, Fall 2006

Prof. David Jerison

**Course Material Related to This Topic:**

- Complete exam problem 1 on page 1
- Check solution to exam problem 1 on page 1
- Complete exam problem 3 on page 1
- Check solution to exam problem 3 on page 1

## Indefinite Integrals

Two questions which involve evaluating indefinite integrals using advanced techniques.

18.01

*Single Variable Calculus*, Fall 2006

Prof. David Jerison

**Course Material Related to This Topic:**

- Complete exam problems 1 to 2 on page 1
- Check solution to exam problems 1 to 2 on page 1

## Evaluating Integrals

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

18.01

*Single Variable Calculus*, Fall 2006

Prof. David Jerison

**Course Material Related to This Topic:**

- Complete exam problem 12 on page 2
- Check solution to exam problem 12 on page 1

## Differentials and Indefinite Integration

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

18.01

*Single Variable Calculus*, Fall 2006

Prof. David Jerison

**Course Material Related to This Topic:**

- Complete exam problems 3A–1 to 3A–3 on page 21
- Check solution to exam problems 3A–1 to 3A–3 on pages 37–9

## Change of Variables and Estimating Integrals

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

18.01

*Single Variable Calculus*, Fall 2006

Prof. David Jerison

**Course Material Related to This Topic:**

- Complete exam problems 3E–1 on page 24 to problems 3E–7 on page 25
- Check solution to exam problems on pages 42–3

## Integration by Direct Substitution

Sixteen integrals to be evaluated using the method of substitution.

18.01

*Single Variable Calculus*, Fall 2006

Prof. David Jerison

**Course Material Related to This Topic:**

- Complete exam problems 5B–1 to 5B–16 on page 36
- Check solution to exam problems 5B–1 to 5B–16 on pages 71–3

## Trigonometric Integrals

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

18.01

*Single Variable Calculus*, Fall 2006

Prof. David Jerison

**Course Material Related to This Topic:**

- Complete exam problems 5C–1 to 5C–14 on page 36
- Check solution to exam problems 5C–1 to 5C–14 on pages 73–4

## Integration by Inverse Substitution

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

18.01

*Single Variable Calculus*, Fall 2006

Prof. David Jerison

**Course Material Related to This Topic:**

- Complete exam problems 5D–1 on page 36 to problems 5D–15 on page 37
- Check solution to exam problems on pages 75–8

## Integration by Partial Fractions

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

18.01

*Single Variable Calculus*, Fall 2006

Prof. David Jerison

**Course Material Related to This Topic:**

- Complete exam problems 5E–1 on page 37 to problems 5E–13 on page 38
- Check solution to exam problems on pages 78–83

## Integration by Parts and Reduction Formulas

Six questions which involve evaluating integrals using the method of integration by parts or deriving reduction formulas.

18.01

*Single Variable Calculus*, Fall 2006

Prof. David Jerison

**Course Material Related to This Topic:**

- Complete exam problems 5F–1 to 5F–6 on page 38
- Check solution to exam problems 5F–1 to 5F–6 on pages 83–5