Integrating trigonometric functions for AQA A-level Maths

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 −sinx.

The integrals of trigonometric expressions involving sec²x, cosec²x, 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 ∫15sin(3x) + 3tan(5x)dx.

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

2

Calculate ∫15x² + 8cos(4x)dx.

∫15x² + 8cos(4x)dx = 5x³ + 2sin(4x) + c, where c is the constant of integration.

3

Integrate the equation of the given graph.

∫10sec²xdx = 10tanx + c, where c is the constant of integration.

4

How can substitution be used to integrate cosecx?

To integrate cosecx, let u = cosecx + cotx and du = (−cosecxcotx − cosec²x)dx.

5

Calculate ∫20x + 12sin(2x)dx.

Using the Chain rule, ∫20x + 12sin(2x)dx = 10x² − 6cos(2x) + c, where c is the constant of integration.

End of page