Question

For $nϵN,$ the value of the definite integral ${\int }_0^{n\pi +V}\sqrt{\frac{1+\cos 2x}{2}}dx$ where $\frac{\pi }{2}<V<\pi$ is

• A. $2n+1-\cos V$
• B. $2n-\sin V$
• C. $2n+2-\sin V$
• D. $2n+1-\sin V$

Answer 1

The correct option is C $2n+2-\sin V$

Given, ${\int }_0^{nx+v}\sqrt{\frac{2{\cos }^{2}x}{2}}dx={\int }_0^{nx+v}\left|\cos x\right|dx$
${\int }_0^{nx}\left|\cos x\right|dx+{\int }_0^{nx+v}\left|\cos x\right|dx$
$n{\int }_0^n\left|\cos x\right|dx+{\int }_0^v\left|\cos x\right|dx$
$2n+{\int }_0^{n/2}\cos xdx-{\int }_{n/2}^v\cos xdx=2n+2-\sin v$