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(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 $

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

Calculus 1 / AB

Chapter 5

Integrals

Section 5

The Substitution Rule

Integration

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University of Michigan - Ann Arbor

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Lectures

03:09

In mathematics, precalculu…

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(a) Show that $ \cos (x^2)…

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01:49

If $f(0)=g(0)=0$ and $f^{\…

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