Evaluate the following definite integrals: _ 0 ^ 2 x^2 dx - Vidyadip

Question

Evaluate the following definite integrals:
$\int_{0}^\limits{\frac{\pi}{2}}\text{x}^2\cos\text{x}\text{ dx}$

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Answer

We have,
$\int\text{x}^2\cos\text{x dx}=\text{x}^2\int\cos\text{x dx}-\int(2\text{x})\big(\int\cos\text{x dx}\big)\text{dx}$
$=\text{x}^2\sin\text{x}-\int\sin\text{x }2\text{x dx}$
$=\text{x}^2\sin\text{x}-2\big[-\text{x}\cos\text{x}+\int\cos\text{x dx}\big]$
$\therefore\ \int_{0}^\limits{\frac{\pi}{2}}\text{x}^2\cos\text{x}\text{ dx}=\big[\text{x}^2\sin\text{x}+2\text{x}\cos\text{x}-2\sin\text{x}\big]^{\frac{\pi}{2}}_0$
$=\Big[\frac{\pi}{4}+0-2-0-0+0\Big]$
$=\frac{\pi^2}{4}-2$

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