Question

If ${\int }_0^{\frac{\mathrm{\pi }}{3}}\frac{\cos x}{3+4\sin x}dx=k\log \left(\frac{3+2\sqrt{3}}{3}\right)$ 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

$\frac{1}{4}$

Explanation for the correct option.

Find the value of $k$:

Given,

${\int }_0^{\frac{\mathrm{\pi }}{3}}\frac{\cos x}{3+4\sin x}dx=k\log \left(\frac{3+2\sqrt{3}}{3}\right)$.

Let, $\sin x=t$

$\Rightarrow \cos xdx=dt$

If, $x=0$then, $t=0$ and $x=\frac{\mathrm{\pi }}{3}$then, $t=\frac{\sqrt{3}}{2}$.

$\begin{array}{rcl}& \Rightarrow & {\int }_0^{\frac{\sqrt{3}}{2}}\frac{1}{3+4t}dt=k\log \left(\frac{3+2\sqrt{3}}{3}\right)\\ & \Rightarrow & \frac{1}{4}{\left[\log 3+4t\right]}_0^{\frac{\sqrt{3}}{2}}=k\log \left(\frac{3+2\sqrt{3}}{3}\right)\\ & \Rightarrow & \frac{1}{4}\left[\log (3+2\sqrt{3})-\log 3\right]=k\log \left(\frac{3+2\sqrt{3}}{3}\right)\\ & \Rightarrow & \frac{1}{4}\left(\log \frac{3+2\sqrt{3})}{3}\right)=k\log \left(\frac{3+2\sqrt{3}}{3}\right)\end{array}$

On comparing the both side, $k=\frac{1}{4}$.

Hence, the correct option is C.