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

Question

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

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Answer

Let $\text{I}=\int_{0}^\limits{\frac{\pi}{2}}\text{x}\cos\text{x}\text{ dx}$ Then,
Integrating by parts
$\text{I}=\big[\text{x}\sin\text{x}\big]^{\frac{\pi}{2}}_0-\int_{0}^\limits{\frac{\pi}{2}}1\sin\text{x}\text{ dx}$
$\Rightarrow\text{I}=\big[\text{x}\sin\text{x}\big]^{\frac{\pi}{2}}_0+\big[\cos\text{x}\big]^{\frac{\pi}{2}}_0$
$\Rightarrow\text{I}=\frac{\pi}{2}-1$

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