# By using the properties of definite integrals,evaluate the integral$\int\limits_\frac{\Large -\pi}{\Large 2}^\frac{\Large \pi}{\Large 2}\sin^7\;x\;dx$

$\begin{array}{1 1} \frac{\pi}{2} \\ 0 \\ \frac{\pi}{4} \\ \pi \end{array}$

Toolbox:
• (i)$\int\limits_a^b f(x)dx=F(b)-F(a)$
• If $f(-x)=-f(x)$ then the function is an odd function. and $\int \limits_{-a}^af(x)=0$
• If $f(-x)=-f(x)$ then the function is an even function. and $\int \limits_{-a}^af(x)=2\int \limits_0^a f(x)$
Given $I=\int\limits_\frac{\Large -\pi}{\Large 2}^\frac{\Large \pi}{\Large 2}\sin^7\;x\;dx$

Now $\sin ^7(-x)=(-\sin x)^7=-\sin^7x$

Hence $\sin^7x$ is an odd function.

Since is an odd function $\int\limits_{-a}^a f(x)=0$

Therefore $\int\limits_\frac{\Large -\pi}{\Large 2}^\frac{\Large \pi}{\Large 2}\sin^7\;x\;dx=0$