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 $\frac{1}{4}$

$\sin x=t\Rightarrow dx.\cos x=dt$

$I={\int }_0^{\frac{\pi }{3}}\frac{dt}{3+4t}$

$=\frac{1}{4}\left[ln\right(3+4t){]}^{\frac{\pi }{3}}$

When $x=0\Rightarrow t=0$

When $x=\frac{\pi }{3}\Rightarrow t=\frac{\sqrt{3}}{2}$

$=\frac{1}{4}\left[ln\right(3+4t){]}^{\frac{\sqrt{3}}{2}}-\frac{1}{4}[ln(3+4t){]}^0$

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

$I=\frac{1}{4}ln\left[\frac{3+2\sqrt{3}}{3}\right]$

$K=\frac{1}{4}$