3.5 Integration

Extend the idea of 'reverse differentiation' to include the integration of e^(ax + b), 1/(ax + b), sin(ax + b), cos(ax + b), sec²(ax + b) and 1/(x² + a²).

Includes examples such as 1/(2 + 3x²).

Taught in: Standard integrals: e, ln, sin, cos, sec² and tan⁻¹ with a linear inside · 6 questions

Use trigonometrical relationships in carrying out integration.

For example, using double-angle formulae to integrate sin²x or cos²(2x).

Taught in: Trig integrals: rewrite with an identity first · 5 questions

Integrate rational functions by decomposing them into partial fractions.

Restricted to the types of partial fractions specified in topic 3.1.

Taught in: Partial fractions in integration: split, then log · 6 questions

Recognise an integrand of the form k f′(x)/f(x), and integrate such functions.

Examples: integrating x/(x² + 1) and tan x.

Taught in: k f′(x) / f(x): the missing numerator trick · 6 questions

Recognise when an integrand can usefully be regarded as a product, and use integration by parts.

Examples: integrating x sin 2x, x²e^(−x), ln x and x tan⁻¹x.

Taught in: Integration by parts: undo the product rule · 6 questions

Use a given substitution to simplify and evaluate either a definite or an indefinite integral.

For example, integrating sin²2x cos x using the substitution u = sin x.

Taught in: Integration by substitution: obey the given u · 6 questions