The value of int_0^pi |sin^3 x| dx is | vedclass

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(C) We need to evaluate the integral $I = \int_0^\pi |\sin^3 x| dx$. Since $\sin x \ge 0$ for all $x \in [0, \pi]$,we have $|\sin^3 x| = \sin^3 x$. Th

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$\frac{4}{3}$

Vedclass · Answer

(C) We need to evaluate the integral $I = \int_0^\pi |\sin^3 x| dx$. Since $\sin x \ge 0$ for all $x \in [0, \pi]$,we have $|\sin^3 x| = \sin^3 x$. Thus,$I = \int_0^\pi \sin^3 x dx$. Using the identity $\sin^3 x = \frac{3 \sin x - \sin 3x}{4}$,we get: $I = \int_0^\pi \frac{3 \sin x - \sin 3x}{4} dx$ $I = \frac{1}{4} [ -3 \cos x + \frac{\cos 3x}{3} ]_0^\pi$ $I = \frac{1}{4} [ (-3 \cos \pi + \frac{\cos 3\pi}{3}) - (-3 \cos 0 + \frac{\cos 0}{3}) ]$ $I = \frac{1}{4} [ (3 - \frac{1}{3}) - (-3 + \frac{1}{3}) ]$ $I = \frac{1}{4} [ \frac{8}{3} - (-\frac{8}{3}) ] = \frac{1}{4} [ \frac{16}{3} ] = \frac{4}{3}$.

The value of $\int_0^\pi |\sin^3 x| dx$ is

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