Question

If ${\int }_0^{\pi /3}\frac{\cos x}{3+4\sin x}dx=k\log \left(\frac{3+2\sqrt{3}}{3}\right)$, then $k$ is equal to

• A. $\frac{1}{2}$
• B. $\frac{1}{3}$
• C. $\frac{1}{4}$
• D. $\frac{1}{8}$

Answer 1

The correct option is C $\frac{1}{4}$

${\int }_0^{\pi /3}\frac{\cos x}{3+4\sin x}dx$

Let $z=3+4\sin x⟹dz=4\cos xdx⟹\cos xdx=\frac{dz}{4}$

When $x=0,z=3$ and when $x=\frac{\pi }{3},z=3+2\sqrt{3}$

Hence, integration becomes:-

$\frac{1}{4}{\int }_3^{3+2\sqrt{3}}\frac{1}{z}dz$

$=\frac{1}{4}{\left[\log z\right]}_3^{3+2\sqrt{3}}$

$=\frac{1}{4}\log \left(\frac{3+2\sqrt{3}}{3}\right)$

$⟹k=\frac{1}{4}$

Hence, answer is option-(C).