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).

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

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

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

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

Solution (PDF) Pages 3 to 4

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

Solution (PDF) Pages 3 to 6

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

Solution (PDF)

Computing an antiderivative using the method of integration by parts.

Solution (PDF) Page 1

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

Solution (PDF) Page 2 to 3

Evaluating an antiderivative that requires the application of multiple techniques.

Solution (PDF) Page 3 to 4

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

Solution (PDF) Page 1

Evaluating an integral using the method of trigonometric substitution.

Solution (PDF) Page 3

Evaluating four integrals using multiple techniques.

Solution (PDF) Pages 4 to 5

Two problems which involve evaluating a definite integral.

Solution (PDF)

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

Solution (PDF)

Two integrals to be evaluated.

Solution (PDF)^# Page 1

Two integrals to be evaluated.

Solution (PDF)^# Page 1

Antidifferentiating a function which is a ratio of polynomials.

Solution (PDF)^# Page 1

Evaluating a definite integral using a suggested trigonometric substitution.

Solution (PDF)^# Page 1

Finding a reduction formula for two integrals involving exponentials.

Solution (PDF)^# Page 1

Evaluating a definite integral using a trigonometric substitution.

Solution (PDF)^# Page 1

Antidifferentiating a function which is a ratio of polynomials.

Solution (PDF)^# Page 1

Two questions which involve evaluating indefinite integrals using advanced techniques.

Solution (PDF)^# Page 1

Evaluating a definite integral using a trigonometric substitution.

Solution (PDF)^# Page 1

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

Solution (PDF)^# Page 1

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

Solution (PDF)^# Page 1

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

Solution (PDF - 4.0 MB)^# Pages 37 to 39

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

Solution (PDF - 4.0 MB)^# Pages 42 to 43

Sixteen integrals to be evaluated using the method of substitution.

Solution (PDF - 4.0 MB)^# Pages 71 to 73

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

Solution (PDF - 4.0 MB) Pages 73 to 74

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

Solution (PDF - 4.0 MB) Pages 75 to 78

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

Solution (PDF - 4.0 MB) Pages 78 to 83

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

Solution (PDF - 4.0 MB) Pages 83 to 85