$\int\limits_{ - 1}^1 {{x^{17}}{{\cos }^4}x} dx = 0$

Let $I = \int\limits_{ - 1}^1 {{x^{17}}{{\cos }^4}x} dx$

Let $f\left( x \right) = {x^{17}}{\cos ^4}x$

$\Rightarrow$ $f\left( { - x} \right) = {\left( { - x} \right)^{17}}{\cos ^4}\left( { - x} \right) = - {x^{17}}{\cos ^4}x = - f\left( x \right)$

$\therefore$ $f\left( x \right)$ is an odd function , hence $I = 0$