allery$\displaystyle\int_0^{\pi/2} \sin^2 x\, dx = ?$
A$\pi/4$
B$\pi/2$
C$1$
D$\pi^2/8$
Answer & Solution
Correct answer: A. $\pi/4$
$\sin^2 x = (1 - \cos 2x)/2$.
$\int_0^{\pi/2} (1 - \cos 2x)/2\, dx = \tfrac{1}{2}[x - \tfrac{1}{2}\sin 2x]_0^{\pi/2} = \tfrac{1}{2}(\pi/2 - 0) = \pi/4$.
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