Edexcel A-Level Mathematics A2 P3.5.2 Integration By Recognition Questions

Practise spotting f′/f logarithmic forms, trig identities and recoverable derivatives before applying bounds or boundary conditions.

Exam points

Recognise f′(x)/f(x) structures and integrate to logarithmic expressions.
Use identities before integrating sin²3x, cos²kx or related trig forms.
Recover f(x) from f′(x) or f″(x), then use boundary or tangent conditions.

[Maximum number: 2]

Find

$\displaystyle \int \sin^2 3x\,dx$

$\cos6x=1-2\sin^2 3x\Rightarrow \sin^2 3x=\frac12-\frac12\cos6x$ $\int\sin^2 3x\,dx=\int\left(\frac12-\frac12\cos6x\right)dx$ $=\frac12x-\frac{1}{12}\sin6x+C$

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Edexcel A-Level Mathematics A2 P3.5.2 Integration By Recognition Questions question 1