Question

Prove: $∫^1_{-1}x^{17}cos^{4}xdx$

Answer 1

Let $I=∫^1_{-1}x^{17}cos^4xdx$
Also,let $ƒ(x)=x^{17}cos^4x$
$⇒ƒ(-x)=(-x)^{17}cos^4(-x)=-x^{17}cos^4x=-ƒ(x)$
Therefore,$ƒ(x)$is an odd function.
It is known that if $f(x)$is an odd function,then$∫^a_{-a}ƒ(x)dx=0$
$∴I=∫^1_{-1}x^{17}cos^4 xdx=0$
Hence,the given result is proved.