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Numerade Educator

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Problem 94 Hard Difficulty

(a) If $ f $ is continuous, prove that
$$ \int^{\pi/2}_0 f(\cos x) \,dx = \int^{\pi/2}_0 f(\sin x) \,dx $$

(b) Use part (a) to evaluate $ \displaystyle \int^{\pi/2}_0 \cos^2 x \,dx $ and $ \displaystyle \int^{\pi/2}_0 \sin^2 x \,dx $

Answer

a. $=\int_{0}^{\pi / 2} f(\sin x) d x$
b. $\int_{0}^{\pi / 2} \cos ^{2} x d x=\int_{0}^{\pi / 2} \sin ^{2} x d x=\frac{\pi}{4}$

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Video Transcript

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