Question

The value of ${\int }_{-\mathrm{\pi }}^{\mathrm{\pi }}\left({\sin }^{3}x\right)\left({\cos }^{2}x\right)dx$ is

• A. $1$
• B. $2$
• C. $3$
• D. $0$

Answer 1

The correct option is D

$0$

Explanation for the correct option:

Compute the required value:

Given: ${\int }_{-\mathrm{\pi }}^{\mathrm{\pi }}\left({\sin }^{3}x\right)\left({\cos }^{2}x\right)dx$

Put $x=-x$

$$\begin{gathered} \Rightarrow {\int }_{-\mathrm{\pi }}^{\mathrm{\pi }}\left({\sin }^3(-x)\right)\left({\cos }^2(-x)\right)dx \\ \Rightarrow -{\int }_{-\mathrm{\pi }}^{\mathrm{\pi }}\left({\sin }^3\left(x\right)\right)\left({\cos }^2\left(x\right)\right)dx \end{gathered}$$

so, ${\int }_{-\mathrm{\pi }}^{\mathrm{\pi }}\left({\sin }^{3}x\right)\left({\cos }^{2}x\right)dx$ is an odd function. $\left[\because {\int }_{-a}^{a}f\left(x\right)dx=0,f(-x)=-f\left(x\right)\right]$

Hence ${\int }_{-\mathrm{\pi }}^{\mathrm{\pi }}\left({\sin }^{3}x\right)\left({\cos }^{2}x\right)dx$ equal to $0$,

Hence, option $D$ is the correct answer.