Evaluate the following integrals: ^ 2 _ - 2 ^3x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}}\sin^3\text{x dx}$

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Answer

Let $\text{I}=\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}}\sin^3\text{x dx}$
$=\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}}\sin\text{x}(1-\cos^2\text{x})\text{dx}$
$=\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}}\sin\text{x}\text{ dx}-\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}}\sin\text{x}\cos^2\text{x dx}$
$=\big[-\cos\text{x}\big]^{\frac{\pi}{2}}_{\frac{-\pi}{2}}+\Big[\frac{\cos^3\text{x}}{3}\Big]^{\frac{\pi}{2}}_{-\frac{\pi}{2}}$
$=0+0=0$

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