Question

The value of $\underset{-1}{\overset{1}{\int }}{\sin }^{11}x.{\cos }^{12}xdx$.

Answer 1

Let $f\left(x\right)={\sin }^{11}x.{\cos }^{12}x$.

Now $f(-x)={\sin }^{11}(-x).{\cos }^{12}(-x)=-{\sin }^{11}x.{\cos }^{12}x=-f\left(x\right)\forall x$.

So $f\left(x\right)$ is an odd function.

Then,

$\underset{-1}{\overset{1}{\int }}{\sin }^{11}x.{\cos }^{12}xdx$

$=0$.[ By property of definite integral]