# 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).

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## Change of Variables

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

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## Review of Inverse Substitution and Another Example

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

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## Antidifferentiating Simple Rational Expressions

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

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## 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).

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## Simplifying Rational Expressions: Partial Fraction Decomposition

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

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## The Heaviside Cover-up Method

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

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

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## Integration by Parts

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

Computing an antiderivative using the method of integration by parts.

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

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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).

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## Reduction Formulas

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

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Finding a reduction formula for two integrals involving exponentials.

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## Advanced Reduction Formulas

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

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## 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.

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## 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.

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## Partial Fraction Decomposition

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

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## 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.

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## 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.

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## Evaluating Definite Integrals

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

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## Miscellaneous Integration Problems

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

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## Partial Fractions

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

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## Techniques of Antidifferentiation

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Evaluating an antiderivative that requires the application of multiple techniques.

Evaluating four integrals using multiple techniques.

## Trigonometric Substitution

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Evaluating an integral using the method of trigonometric substitution.

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Evaluating a definite integral using a suggested trigonometric substitution.

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

## Integration by Substitution

Two problems which involve evaluating a definite integral.

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## Antiderivatives of Inverse Trigonometric Functions

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

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## Definite Integrals

Two integrals to be evaluated.

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## Indefinite Integrals: Ratio of Polynomials

Antidifferentiating a function which is a ratio of polynomials.

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## Indefinite Integrals

Two questions which involve evaluating indefinite integrals using advanced techniques.

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## Evaluating Integrals

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

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## Differentials and Indefinite Integration

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

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## Change of Variables and Estimating Integrals

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

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## Integration by Direct Substitution

Sixteen integrals to be evaluated using the method of substitution.

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## Trigonometric Integrals

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

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## Integration by Inverse Substitution

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

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## Integration by Partial Fractions

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

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## Integration by Parts and Reduction Formulas

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

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