Question

The value of $\underset{-\mathrm{\pi }}{\overset{\mathrm{\pi }}{\int }}{\sin }^{3}x{\cos }^{2}xdx$ is _______________.

Answer 1

$$\begin{gathered} \mathrm{Let}I=\underset{-\mathrm{\pi }}{\overset{\mathrm{\pi }}{\int }}{\sin }^{3}x{\cos }^{2}xdx \\ \mathrm{Let}, \\ f\left(x\right)={\sin }^{3}x{\cos }^{2}x \\ f\left(-x\right)={\sin }^3\left(-x\right){\cos }^2\left(-x\right) \\ =-{\sin }^{3}x{\cos }^{2}x \\ =-f\left(x\right) \\ \mathrm{We}\mathrm{know}, \\ \underset{-a}{\overset{a}{\int }}f\left(x\right)dx=\left\{\begin{array}{l}0,\mathrm{if}f\left(-x\right)=-f\left(x\right)\\ 2\underset{0}{\overset{a}{\int }}f\left(x\right)dx,\mathrm{if}f\left(-x\right)=f\left(x\right)\end{array}\right. \\ \mathrm{Thus}, \\ \underset{-\mathrm{\pi }}{\overset{\mathrm{\pi }}{\int }}{\sin }^{3}x{\cos }^{2}xdx=0 \end{gathered}$$

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Hence, the value of $\underset{-\mathrm{\pi }}{\overset{\mathrm{\pi }}{\int }}{\sin }^{3}x{\cos }^{2}xdx$ is 0.