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

Question

Evaluate the following definite integrals:
$\int_{0}^\limits{\frac{\pi}{2}}\sin\text{x }\sin2\text{x}\text{ dx}$

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Answer

Let $\text{I}=\int_{0}^\limits{\frac{\pi}{2}}\sin\text{x }\sin2\text{x}\text{ dx}$ Then,
$\text{I}=\int_{0}^\limits{\frac{\pi}{2}}2\sin^2\text{x }\cos\text{x}\text{ dx}$
$\Rightarrow\text{I}=\int_{0}^\limits{\frac{\pi}{2}}2(1-\cos^2\text{x})\cos\text{x}\text{ dx}$
$\Rightarrow\text{I}=\int_{0}^\limits{\frac{\pi}{2}}\big(2\cos\text{x}-2\cos^3\text{x}\big)\text{dx}$
$\Rightarrow\text{I}=\bigg[2\sin\text{x}-2\Big(\sin\text{x}-\frac{\sin^3\text{x}}{3}\Big)\bigg]^{\frac{\pi}{2}}_0$
$\Rightarrow\text{I}=\Big[2-2\Big(1-\frac{1}{3}\Big)\Big]-0$
$\Rightarrow\text{I}=\frac{2}{3}$

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