Question

${\displaystyle {\int }_0^{\frac{\pi }{2}}{\cos }^{2}x dx}$ = ______ 

Options

• ${\displaystyle \frac{\pi }{4}}$

• ${\displaystyle \frac{\pi }{3}}$

• ${\displaystyle \frac{\pi }{2}}$

• π

Answer 1

${\displaystyle {\int }_0^{\frac{\pi }{2}}{\cos }^{2}x dx}$ = ${\displaystyle \underline{\frac{\pi }{4}}}$

Explanation:

${\displaystyle {\int }_0^{\frac{\pi }{2}}{\cos }^{2}x dx={\int }_0^{\frac{\pi }{2}}\left(\frac{1+\cos 2x}{2}\right)dx}$

= ${\displaystyle \frac{1}{2}[{\int }_0^{\frac{\pi }{2}}1dx+{\int }_0^{\frac{\pi }{2}}\cos 2xdx]}$

= ${\displaystyle \frac{1}{2}[{\left[x\right]}_0^{\frac{\pi }{2}}+{\left[\frac{\sin 2x}{2}\right]}_0^{\frac{\pi }{2}}]}$

= ${\displaystyle \frac{1}{2}[(\frac{\pi }{2}-0)+(\frac{1}{2}\sin \pi -0)]=\frac{\pi }{4}}$