By Using properties of definite integral, evaluate the following … - Vidyadip

Question

By Using properties of definite integral, evaluate the following integral in Exercise:
$\int^{2\pi}_{0}\cos^{5}\text{x}\ \text{dx}$

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Answer

$\int^{2\pi}\limits_{0}\cos^{5}\ \text{x}\ \text{dx}=2\int^{\pi}\limits_{0}\cos^{5}\text{x}\ \bigg[\because\int^{2\text{a}}\limits_{0}\text{f}\text{(x)}\text{dx}=2\int^{\text{a}}\limits_{0}\text{f}\text{(x)}\text{dx},\text{if}\ \text{f}(2\text{a}-\text{x})=\text{f}\text{(x)}\bigg]$
$\text{Here}\ \text{f}\text{(x)}=\cos^{5}\text{x}\ \therefore\ \text{f}(2\pi-\text{x}=\cos^{5}(2\pi-\text{x})=\cos^{5}\text{x}$
$\Rightarrow\ \ \text{f}\ \text{(x)}=2(0)=0$

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