Study

4 pages

Chapter 7: Integration

Reference: Pearson Edexcel International A Level Pure Mathematics 3 Student Book. This chapter extends integration to standard functions, trig identities and reverse chain rule forms.

What you'll learn

Main skills from this lesson

Integrate standard exponential, logarithmic and trigonometric forms

Use simple reverse chain rule patterns

Choose identities to make integrals easier

Page 1

Standard Integrals

Integration reverses differentiation. Keep these close:

integral e^x dx = e^x + C

integral cos x dx = sin x + C

integral sin x dx = -cos x + C

integral 1/x dx = ln|x| + C

Check by differentiating your answer.

Remember

For 1/x, use ln|x| because x may be negative in an interval.

Page 2

Integrating f(ax + b)

When the inside is linear, divide by the coefficient of x.

Example pattern:

integral cos(3x + 2) dx = (1/3)sin(3x + 2) + C

This is reverse chain rule in its friendliest form.

inside derivative = 3, so divide by 3

Page 3

Using Trig Identities

Some trig integrals are easier after rewriting.

For powers of sin and cos, use identities such as:

cos 2x = 1 - 2sin^2 x

cos 2x = 2cos^2 x - 1

The goal is to turn the expression into standard integrals.

Page 4

Study Tips for Integration

Integration is a pattern-recognition topic.

Make a formula sheet from memory, then check it.

For every integral, ask: is this standard, reverse chain, or identity first?

Differentiate your answer to self-mark.

Use brackets generously around ax + b.

Practice

Integrate 5e^(5x).

e^(5x) + C, because differentiating e^(5x) gives 5e^(5x).

Great work

Continue with quizzes, flashcards, or games when you are ready.

Pure Mathematics 3 chapters