Question

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

• A. $2$
• B. $1$
• C. $\frac{1}{2}$
• D. $0$

Answer 1

The correct option is B $1$

${\int }_0^{\frac{\pi }{2}}\frac{\cos 3x+1}{2\cos x-1}dx$

$={\int }_0^{\frac{\pi }{2}}\frac{4{\cos }^{3}x-3\cos x+1}{2\cos x-1}dx$

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

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

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

$={\sin x+\frac{1}{2}\sin 2x]}_0^{\frac{\pi }{2}}$

$=1$.