Question

If${\int }_0^{\pi /3}\frac{\cos }{3+4\sin x}dx=K\log \frac{(3+2\sqrt{3})}{3}$ then K is

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

Answer 1

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

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

Substituting $t=3+4\sin x\Rightarrow dt=4\cos x$

$I=\frac{1}{4}{\int }_3^{3+2\sqrt{3}}\frac{1}{t}dt={\left[\frac{\log t}{4}\right]}_3^{3+2\sqrt{3}}$

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