Question

The value of ${\int }_0^{\pi /2}\frac{\cos 3x+1}{2\cos x-1}dx$ is

• A. 2
• B. 1
• C. 1/2
• D. 0

Answer 1

The correct option is B 1

Given , ${\int }_0^{\pi /2}\frac{\cos 3x+1}{2\cos x-1}dx$
$=$ ${\int }_0^{\pi /2}\frac{4{\cos }^{3}x-3\cos x+1}{2\cos x-1}dx$
$=$ ${\int }_0^{\pi /2}(\frac{2{\cos }^{2}x(2\cos x-1)}{2\cos x-1}+\frac{\cos x(2\cos x-1)}{2\cos x-1}-\frac{(2\cos x-1)}{2\cos x-1})dx$
$=$ ${\int }_0^{\pi /2}(2{\cos }^{2}x+\cos x-1)dx$
$=$ ${\int }_0^{\pi /2}(\cos 2x+\cos x)dx$
$=$ $\frac{\sin 2\left(\frac{\pi }{2}\right)}{2}+\sin \frac{\pi }{2}-0=1$