# Evaluate : $\int_{-\Large\frac{\pi}{2}}^{\Large\frac{\pi}{2}} \sin^7x\: dx$

Toolbox:
• $\int\limits_a^b f(x)dx=F(b)-F(a)$
• If $f(-x)=-f(x)$ then the function is an odd function.
• $\Rightarrow \int \limits_{-a}^af(x)=0$
• If $f(-x)=-f(x)$ then the function is an even function.
• $\Rightarrow \int \limits_{-a}^af(x)=2\int \limits_0^a f(x)$
Step 1:
$I=\int _{\Large\frac{-\pi}{2}}^{\Large\frac{\pi}{2}}\sin^7xdx$
Let $f(x)=\sin^7x$
$f(-x)=\sin^7(-x)$
$\qquad\;\;=-\sin^7x$
$\therefore f(-x)=-f(x)$
Hence it is an odd function.
Step 2:
$\therefore\int \limits_{-a}^af(x)=0$,if $f(x)$ is an odd function.
$\Rightarrow \int _{\Large\frac{-\pi}{2}}^{\Large\frac{\pi}{2}}\sin^7xdx=0$
Because $\sin^7(x)$ is an odd function.