Integration

Maths Pure 3 · Integration. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

Test this topic free All Maths Pure 3 topics

The idea

Integration reverses differentiation, so each standard derivative gives a standard integral. When the function contains a linear expression (ax + b), integrate as usual and then divide by a.

Many integrals must be rewritten first. Use a trigonometric identity for sin² x, cos² x or tan² x. Use partial fractions for a rational function. If the top is the derivative of the bottom, the answer is the natural logarithm of the bottom.

Two further methods remain. Substitution replaces part of the integrand by u; you must also change dx and the limits. Integration by parts is for a product such as x e^x or x ln x: you differentiate one part and integrate the other.

Rules to remember

∫ e^{ax + b} dx = (1/a)e^{ax + b} + c; ∫ 1/(ax + b) dx = (1/a) ln|ax + b| + c
∫ sin(ax + b) dx = −(1/a) cos(ax + b) + c; ∫ cos(ax + b) dx = (1/a) sin(ax + b) + c
∫ sec²(ax + b) dx = (1/a) tan(ax + b) + c
∫ 1/(x² + a²) dx = (1/a) tan⁻¹(x/a) + c
sin² x = ½(1 − cos 2x); cos² x = ½(1 + cos 2x); tan² x = sec² x − 1
∫ f′(x)/f(x) dx = ln|f(x)| + c
By parts: ∫ u(dv/dx) dx = uv − ∫ v(du/dx) dx; choose u = ln x when ln x appears, otherwise u = the power of x

Common mistake

Students multiply by a instead of dividing, for example writing ∫ cos 3x dx = 3 sin 3x. Integration divides: the answer is (1/3) sin 3x + c. Differentiate your answer to check that it gives back the integrand.

Worked example

Find the exact value of ∫₀^π/2 x sin x dx.

By parts with u = x and dv/dx = sin x, so du/dx = 1 and v = −cos x.
∫ x sin x dx = −x cos x + ∫ cos x dx = −x cos x + sin x.
At x = π/2: −(π/2)(0) + 1 = 1. At x = 0: 0 + 0 = 0.
Value = 1 − 0 = 1.

Answer: 1

Ask about this

Type what you are stuck on. The helper finds the explanation, worked examples and the notes to read, free and instant.

Read the full notes

Keep going

← Differentiation Numerical solution of equations →Take a Maths Pure 3 test