Question

If $ \int\limits_{0}^{\frac{\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

Answer: (C)

We have,
$\int\limits_{0}^{\pi /3} \frac{\cos\,x}{3+4\,\sin\,x} dx $
$=k\,\log \left(\frac{3+2\sqrt{3}}{3}\right)\ldots\left(i\right)$
Let $ I=\int\limits_{0}^{\pi/ 3} \frac{\cos\,x}{3+4\,\sin\,x}dx $
Put $3 + 4 \sin \,x = t $
$\Rightarrow 0+4\,\cos\,x\,dx =dt$
Upper limit, $x=\frac{\pi}{3}, t=3+4\,\sin \frac{\pi}{3}$
$=3+4\times\frac{\sqrt{3}}{2}=3+2\sqrt{3}$
and lower limit $x=0$,
$t=3+4\,\sin \, 0=3 $
$\therefore I=\int\limits_{3}^{3+2\sqrt{3}} \frac{dt}{4t}=\frac{1}{4} \left[\log\,t\right]_{3}^{3+2\sqrt{3}}$
$=\frac{1}{4}\left[\log\left(3+2\sqrt{3}\right)-\log\,3\right]$
$\Rightarrow I=\frac{1}{4} \log \left(\frac{3+2\sqrt{3}}{3}\right) \ldots\left(ii\right)$
$\therefore $ From Eqs. $\left(i\right)$ and $\left(ii\right)$, we get
$k=\frac{1}{4}$