Question

State true or false:

The average value of the function $f\left(x\right)=si{n}^{2}xco{s}^{3}x$ on the interval $[-\pi ,\pi ]$ is 0.

• A. True
• B. False

Answer 1

The correct option is A True

${\int }_{-\pi }^{\pi }f\left(x\right)dx={\int }_{-\pi }^{\pi }{\sin }^{2}x{\cos }^{3}xdx={\int }_{-\pi }^{\pi }{\sin }^{2}x(1-{\sin }^{2}x)\cos xdx={\int }_{-\pi }^{\pi }{\sin }^{2}x\cos xdx-{\int }_{-\pi }^{\pi }{\sin }^{4}x\cos xdx$
$={\left[\frac{{\sin }^{3}x}{3}\right]}_{-\pi }^{\pi }-{\left[\frac{{\sin }^{5}x}{3}\right]}_{-\pi }^{\pi }=0$