Substitution u = sin x or u = cos x (area/integral)

Evaluate a definite integral or area using substitution u = sin x, u = cos x, or u = f(trig x) where the substitution is into a product/composition of trig functions, typically involving powers of sin and cos.

3 questions · Standard +0.8

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Edexcel AEA 2012 June Q2

10 marks Challenging +1.8

2.(a)Show that $$\sin 3 x = 3 \sin x - 4 \sin ^ { 3 } x$$ Hence find
(b) \(\int \cos x ( 6 \sin x - 2 \sin 3 x ) ^ { \frac { 2 } { 3 } } \mathrm {~d} x\) (c) \(\int ( 3 \sin 2 x - 2 \sin 3 x \cos x ) ^ { \frac { 1 } { 3 } } \mathrm {~d} x\)

CAIE P3 2020 Specimen Q9

10 marks Standard +0.3

9 \includegraphics[max width=\textwidth, alt={}, center]{c1eee696-3d7f-410a-91a8-fa902309c117-16_307_593_269_735} The diagram shows the curve \(y = \sin ^ { 2 } 2 x \cos x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\).

Find the \(x\)-coordinate of \(M\).
Using the substitution \(u = \sin x\), find the area of the shaded region bounded by the curve and the \(x\)-axis.

OCR C4 2012 January Q5

6 marks Standard +0.3

5 Use the substitution \(u = \cos x\) to find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \sin ^ { 3 } x \cos ^ { 2 } x d x$$