Question

Solve: $\underset{-\frac{\pi }{2}}{\overset{\frac{\pi }{2}}{\int }}{\sin }^{5}x.{\cos }^{100}xdx$

Answer 1

We have ${\sin }^{5}x.{\cos }^{100}x$ is an odd function.

As ${\sin }^5(-x).{\cos }^{100}(-x)$$=-{\sin }^{5}x.{\cos }^{100}x$

So,

$\underset{-\frac{\pi }{2}}{\overset{\frac{\pi }{2}}{\int }}{\sin }^{5}x.{\cos }^{100}xdx=0$ [ Using property of definite integral]