Question

By using the properties of definite integrals, evaluate the integral ${\int }_0^{2\pi }{\cos }^{5}xdx$

Answer 1

Let $I={\int }_0^{2\pi }{\cos }^{5}xdx$ ........... (1)
We know that, ${\cos }^5(2\pi -x)={\cos }^{5}x$
Also It is known that,
${\int }_0^{2a}f\left(x\right)dx=\{\begin{array}{c}2{\int }_0^{a}f\left(x\right)dx,\text{if}f(2a-x)=f\left(x\right)\\ =0\text{if}f(2a-x)=-f\left(x\right)\end{array}$
$\therefore I={\int }_0^{2\pi }{\cos }^{5}xdx$
$\Rightarrow I=2{\int }_0^{\pi }{\cos }^{5}xdx=2\left(0\right)=0,[\because {\cos }^5(\pi -x)=-{\cos }^{5}x]$