Edexcel IAL Mathematics P3.5.2 integration by recognition

The direct questions reward spotting the structure first: f′(x)/f(x) gives logarithms, trig identities simplify powers, and constants come from limits or boundary data.

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: 4]

Hence, using algebraic integration, find the exact value of

\int_{\frac{\pi}{12}}^{\frac{\pi}{8}}\left(5-4 \cos ^{2} 3 x\right) \mathrm{d} x

M1 A1

\int(3-2\cos 6x)\,dx=3x-\frac{\sin 6x}{3}

dM1 A1

\left[3x-\frac{\sin 6x}{3}\right]_{\pi/12}^{\pi/8}=\frac{\pi}{8}+\frac{2-\sqrt{2}}{6}

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P3.5.2 - Integration by recognition question 1