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Q.
The value of the integral $\displaystyle \int _{- 3 \pi }^{3 \pi } \left|s i n^{3} x\right|dx$ is equal to
NTA AbhyasNTA Abhyas 2020Integrals
Solution:
The period of the function $\left|\sin ^{3} x\right|$ is $\pi,$ thus $I=6 \int_{0}^{\pi} \sin ^{3} x d x$ $\Rightarrow I=6 \int_{0}^{\pi} \frac{3 \sin x-\sin (3 x)}{4} d x\left(\right.$ As $\left.\left(\sin (3 x)=3 \sin x-4 \sin ^{3} x\right)\right)$
$\Rightarrow I =\frac{6}{4} \int_{0}^{\pi}(3 \sin x-\sin (3 x)) d x$
$\Rightarrow I=\frac{3}{2}\left([-3 \cos x]_{0}^{\pi}+\left[\frac{\cos (3 x)}{3}\right]_{0}^{\pi}\right)$
$=\frac{3}{2}\left(3-(-3)+\left(\frac{-1}{3}-\frac{1}{3}\right)\right)$
$=\frac{3}{2}\left(6-\frac{2}{3}\right)=\frac{3}{2} \times \frac{16}{3}=8$