Question

The value of $\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\cos }^{2}xdx$ is

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

Answer 1

The correct option is B $\frac{\pi }{4}$

Let $I=\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\cos }^{2}xdx$
we know that,
$\cos 2x=2{\cos }^{2}x-1$
$\Rightarrow {\cos }^{2}x=\frac{1+\cos 2x}{2}$
$\Rightarrow I=\underset{0}{\overset{\pi /2}{\int }}\left(\frac{1+\cos 2x}{2}\right)dx=\frac{1}{2}\underset{0}{\overset{\frac{\pi }{2}}{\int }}dx+\frac{1}{2}\underset{0}{\overset{\frac{\pi }{2}}{\int }}\cos 2xdx$
$=\frac{1}{2}{\left[x\right]}_0^{\frac{\pi }{2}}+\frac{1}{2}{\left[\frac{\sin 2x}{2}\right]}_0^{\frac{\pi }{2}}=\frac{1}{2}(\frac{\pi }{2}-0)+\frac{1}{4}(0-0)=\frac{\pi }{4}$