If-f-x-is-an-even-function-then-0-pi-f-cos-x-dx-2-0-pi-2-f-cos-x-dx-

Question Number 84861 by Jakir Sarif Mondal last updated on 16/Mar/20

If f (x) is an even function, then

\underset{0}{\int^{π}} f (\cos x) d x = 2 \underset{0}{\int^{π / 2}} f (\cos x) d x

Commented by mr W last updated on 17/Mar/20

f (x) i s e v e n, \Rightarrow f (- x) = f (x)

\int_{0}^{π} f (\cos x) d x

= \int_{0}^{\frac{π}{2}} f (\cos x) d x + \int_{\frac{π}{2}}^{π} f (\cos x) d x

= I_{1} + I_{2}

l e t x = t + \frac{π}{2}

\Rightarrow t = x - \frac{π}{2}

d x = d t

\frac{π}{2} ⩽ x ⩽ π \Rightarrow 0 ⩽ t ⩽ \frac{π}{2}

\cos x = \cos (t + \frac{π}{2}) = - \cos t

f (\cos x) = f (- \cos t) = f (\cos t)

I_{2} = \int_{\frac{π}{2}}^{π} f (\cos x) d x

= \int_{0}^{\frac{π}{2}} f (\cos t) d t

= I_{1}

\Rightarrow \int_{0}^{π} f (\cos x) d x = 2 I_{1} = 2 \int_{0}^{\frac{π}{2}} f (\cos x) d x

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