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Integrating trigonometric functions for AQA A-level Maths

Integrating trigonometric functions

This page covers the following topics:

1. Integrating simple trigonometric functions
2. Integrating further trigonometric functions
3. Integrating trigonometric functions with substitution
4. Solving integration problems with trigonometric functions

The integral of sinx is cosx and the integral of cosx is โˆ’sinx.

Integrating simple trigonometric functions

The integrals of trigonometric expressions involving secยฒx, cosecยฒx, secxtanx and cosecxcotx can be found using the integral results given.

Integrating further trigonometric functions

The integrals of trigonometric functions can be found using substitution.

Integrating trigonometric functions with substitution

All integration results can be used in combination to solve integration problems.

Solving integration problems with trigonometric functions

1

Solve โˆซ15sin(3x) + 3tan(5x)dx.

โˆซ15sin(3x) + 3tan(15x)dx = โˆ’5cos(3x) โˆ’ (3/15)ln|cos(15x)| + c = โˆ’5cos(3x) โˆ’ (1/5)ln|cos(15x)| + c, where c is the constant of integration.

Solve โˆซ15sin(3x) + 3tan(5x)dx.

2

Calculate โˆซ15xยฒ + 8cos(4x)dx.

โˆซ15xยฒ + 8cos(4x)dx = 5xยณ + 2sin(4x) + c, where c is the constant of integration.

Calculate โˆซ15xยฒ + 8cos(4x)dx.

3

Integrate the equation of the given graph.

โˆซ10secยฒxdx = 10tanx + c, where c is the constant of integration.

Integrate the equation of the given graph.

4

How can substitution be used to integrate cosecx?

To integrate cosecx, let u = cosecx + cotx and du = (โˆ’cosecxcotx โˆ’ cosecยฒx)dx.

How can substitution be used to integrate cosecx?

5

Calculate โˆซ20x + 12sin(2x)dx.

Using the Chain rule, โˆซ20x + 12sin(2x)dx = 10xยฒ โˆ’ 6cos(2x) + c, where c is the constant of integration.

Calculate โˆซ20x + 12sin(2x)dx.

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