The value of int limits_{0}^{pi}|cos x|^{3} d | vedclass

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Question

(B) Let $I = \int \limits_{0}^{\pi} |\cos x|^3 dx$. Since the function $f(x) = |\cos x|^3$ is symmetric about $x = \frac{\pi}{2}$ in the interval $[0,

Vedclass · Accepted Answer

$\frac{4}{3}$

Vedclass · Answer

(B) Let $I = \int \limits_{0}^{\pi} |\cos x|^3 dx$. Since the function $f(x) = |\cos x|^3$ is symmetric about $x = \frac{\pi}{2}$ in the interval $[0, \pi]$,we can write: $I = 2 \int \limits_{0}^{\frac{\pi}{2}} \cos^3 x dx$. Using Wallis' formula,$\int \limits_{0}^{\frac{\pi}{2}} \cos^n x dx = \frac{(n-1)!!}{n!!}$ for odd $n$: $I = 2 	imes \left( \frac{3-1}{3} ight) = 2 	imes \frac{2}{3} = \frac{4}{3}$. Thus,the value is $\frac{4}{3}$.

The value of $\int \limits_{0}^{\pi}|\cos x|^{3} dx$ is

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