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

Question

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

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Answer

$\text{Let}\text{ I}=\int\limits^\frac{\pi}{2}_{-\frac{\pi}{2}}\sin^3\text{x}\text{ dx}$
$=\int\limits^\frac{\pi}{2}_{-\frac{\pi}{2}}\sin\text{x}\sin^2\text{x}\text{ dx}$
$=\int\limits^\frac{\pi}{2}_{-\frac{\pi}{2}}\sin\text{x}(1-\cos^2\text{x})\text{dx}$
$\text{Let }\cos\text{x} = \text{t}, \text{then}-\sin\text{x}\text{ dx}=\text{dt}$
$\text{when}, \text{x}\rightarrow-\frac{\pi}{2};\text{t}\rightarrow0\text{ and }\text{x}\rightarrow\frac{\pi}{2};\text{t}\rightarrow0$
$\text{I}=\int\limits^0_0(-1+\text{t}^2)\text{dt}$
$= 0$

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