Question

Prove ${\int }_{-1}^1{x}^{17}{\cos }^{4}xdx=0$

Answer 1

Let $I={\int }_{-1}^1{x}^{17}{\cos }^{4}xdx$
Also, let $f\left(x\right)=x^{17}{\cos }^{4}x$
$\Rightarrow f(-x)=(-x{)}^{17}{\cos }^4(-x)=-x^{17}{\cos }^{4}x=-f(x)$
Therefore, $f\left(x\right)$ is an odd function.
It is known that if $f\left(x\right)$ is an odd function, then ${\int }_{-a}^{a}f\left(x\right)dx=0$
$\therefore I={\int }_{-1}^1{x}^{17}{\cos }^{4}xdx=0$
Hence, the given result is proved.