Question

$\underset{0}{\overset{\mathrm{\pi }/2}{\int }}{\cos }^{2}xdx$

Answer 1

$$\begin{gathered} \mathrm{Let}I={\int }_0^{\frac{\mathrm{\pi }}{2}}{\cos }^{2}x\mathit{d}x.\mathrm{Then}, \\ I={\int }_0^{\frac{\mathrm{\pi }}{2}}{\cos }^{2}x\mathit{d}x \\ \Rightarrow I=\frac{1}{2}{\int }_0^{\frac{\mathrm{\pi }}{2}}\left(1+\cos 2x\right)dx\left[\because \cos 2x=2{\cos }^{2}x-1\right] \\ \Rightarrow I={\left[\frac{x}{2}+\frac{\sin 2x}{4}\right]}_0^{\frac{\mathrm{\pi }}{2}} \\ \Rightarrow I=\frac{\mathrm{\pi }}{4}+0-0 \\ \Rightarrow I=\frac{\mathrm{\pi }}{4} \end{gathered}$$