Question

Evaluate : ${\int }_{-\pi /4}^{\pi /4}{x}^5{\cos }^{2}xdx$.

Answer 1

Let $I={\int }_{\frac{-\pi }{4}}^{+\pi /4}{x}^5{\cos }^{2}xdx$

Using the property of definite integral

${\int }_{-a}^{+a}f\left(x\right)dx={\int }_0^a\left[f\right(x)+f(-x\left)\right]dx$

$I={\int }_0^{\pi /4}{x}^5{\cos }^{2}x+(-x{)}^5{\cos }^2(-x)dx$

$I={\int }_0^{\pi /4}{\cos }^{2}x(x^5-x^5)dx=0$

Hence the answer is 0