Integration by Substitution, Parts & Partial Fractions

Antidifferentiation: Integration by Substitution

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).

• 18.01 Single Variable Calculus, Fall 2005
  Prof. Jason Starr

Course Material Related to This Topic:

• Read lecture notes, section 4 on pages 3–4

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:

• Read lecture notes, section 5 on pages 4–5

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:

• Read lecture notes, section 1 on pages 1–4

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:

• Read lecture notes, section 2 on pages 4–5

Simplifying Rational Expressions: Division and Factoring

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

• 18.01 Single Variable Calculus, Fall 2005
  Prof. Jason Starr

Course Material Related to This Topic:

• Read lecture notes, section 3 on pages 5–6

Simplifying Rational Expressions: Partial Fraction Decomposition

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

• 18.01 Single Variable Calculus, Fall 2005
  Prof. Jason Starr

Course Material Related to This Topic:

• Read lecture notes, section 4 on pages 6–7

The Heaviside Cover-up Method

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

• 18.01 Single Variable Calculus, Fall 2005
  Prof. Jason Starr

Course Material Related to This Topic:

• Read lecture notes, section 5 on pages 7–9

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:

• Read lecture notes, section on pages 1–3

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).

• Read lecture notes, section 1 on pages 1–2

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.

• Read chapter 19 of online textbook

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:

• Read lecture notes, section 2 on pages 2–3

Reduction Formulas

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

• 18.01 Single Variable Calculus, Fall 2005
  Prof. Jason Starr

Course Material Related to This Topic:

• Read lecture notes, section 2 on pages 3–4

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:

• Read lecture notes, section 3 on pages 4–5

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:

• Read lecture notes, section 3 on pages 3–4

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:

• Read chapter 19 of online textbook

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:

• Read chapter 19 of online textbook

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:

• Read chapter 27 of online textbook

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