Evaluate the following definite integrals: _ 0 ^ 2 ^3x dx - Vidyadip

Question

Evaluate the following definite integrals:
$\int\limits_{0}^{\frac{\pi}{2}}\cos^3\text{x}\text{ dx}$

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Answer

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

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