Question

${\int }_0^{\pi /3}\frac{\cos \theta }{3+4\sin \theta }d\theta =\lambda \log \frac{3+2\sqrt{3}}{3}$ then $\lambda$ equals

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

Answer 1

Answer: (C)

$\frac{1}{4}$

Given ${\int }_0^{\pi /3}\frac{\cos \theta }{3+4\sin \theta }d\theta =\lambda \log \frac{3+2\sqrt{3}}{3}$ ....(1)
Consider, $I={\int }_0^{\pi /3}\frac{\cos \theta }{3+4\sin \theta }d\theta$
Substitute $\sin \theta =t$
$\Rightarrow \cos \theta d\theta =dt$
$I={\int }_0^{\sqrt{3}/2}\frac{dt}{3+4t}$
Substitute $3+4t=u$
$4dt=du$
$\therefore I=\frac{1}{4}{\int }_3^{3+2\sqrt{3}}\frac{du}{u}$
$=\frac{1}{4}[\log u{]}_3^{3+2\sqrt{3}}$
$I=\frac{1}{4}\left[\log (3+2\sqrt{3})-\log 3\right]$
$\Rightarrow I=\frac{1}{4}\log \frac{3+2\sqrt{3}}{3}$
So, on comparing with (1), we get
$\lambda =\frac{1}{4}$