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A-level Maths Edexcel Integration
Evaluate $\displaystyle\int_0^{\pi/2} \sin^2 x \, dx$
Model answer
Use the double-angle identity $\cos 2x \equiv 1 - 2\sin^2 x$, so $\sin^2 x = \dfrac{1 - \cos 2x}{2}$.

$$\int_0^{\pi/2} \sin^2 x \, dx = \int_0^{\pi/2} \frac{1 - \cos 2x}{2} \, dx = \left[\frac{x}{2} - \frac{\sin 2x}{4}\right]_0^{\pi/2}$$$$= \left(\frac{\pi}{4} - \frac{\sin \pi}{4}\right) - \left(0 - \frac{\sin 0}{4}\right) = \frac{\pi}{4}$$
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IGCSE Biology · CAIE
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