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INTEGRALS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINTEGRALS2 Marks
Question
By Using properties of definite integral, evaluate the following integral in Exercise:
$\int^{\frac{\pi}{2}}_{\frac{-\pi}{2}}\sin^{7}\text{x}\ \text{dx}$
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Answer

$\text{Let}\ \text{I}=\int^{\frac{\pi}{2}}\limits_{\frac{-\pi}{2}}\sin^{7}\text{x}\ \text{dx}$
$\text{Here}\ \ \text{f}\text{(x)}=\sin^{7}\text{x}$
$\therefore\ \ \text{f}(-\text{x)}=\sin^{7}(-\text{x})=(-\sin^{7}\text{x})=-\sin^{7}\text{x}=-\text{f}\text{(x)}$
$\therefore\ \ \text{f}(\text{x})\ \text{is an odd function of x.}$
$\therefore\ \ \text{I}=\int^{\frac{\pi}{2}}\limits_{\frac{-\pi}{2}}\sin^{7}\text{x}\ \text{dx}=0\ \ \bigg[\because\int^{\text{a}}\limits_{-\text{a}}\text{f}\text{(x)}\text{dx}=0,$when $\text{f(x)}$ is an odd function$\bigg]$

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