Question

The integral ${\int }_0^{\mathrm{\pi }}{\left|\cos x\right|}^{3}dx$ is equal to;

• A. $\frac{4}{3}$
• B. $\frac{2}{3}$
• C. $0$
• D. $\frac{8}{3}$

Answer 1

The correct option is A

$\frac{4}{3}$

Explanation for the correct option:

$$\begin{gathered} {\int }_0^{\mathrm{\pi }}{\left|\cos x\right|}^{3}dx=2{\int }_0^{\frac{\mathrm{\pi }}{2}}{\cos }^{3}x·dx \\ =2{\int }_0^{\frac{\mathrm{\pi }}{2}}\frac{3\cos x+\cos 3x}{4} \\ =\frac{1}{2}{{\left(3\sin x+\frac{\sin 3x}{3}\right)}_0}^{\frac{\mathrm{\pi }}{2}} \\ =\frac{1}{2}\left(3-\frac{1}{3}\right) \\ =2\times \frac{8}{3} \\ =\frac{4}{3} \end{gathered}$$

Hence, option (A) is correct.