Question

${\int }_0^{\pi /3}\frac{\cos \theta }{5-4\sin \theta }d\theta$ equal to.

• A. $\frac{1}{4}log\left(\frac{5}{5+2\sqrt{3}}\right)$
• B. $\frac{1}{4}log\left(\frac{5}{5-2\sqrt{3}}\right)$
• C. $\frac{1}{4}log\left(\frac{5+2\sqrt{3}}{5}\right)$
• D. $\frac{1}{4}log\left(\frac{5-2\sqrt{3}}{5}\right)$

Answer 1

Answer: (B)

$\frac{1}{4}log\left(\frac{5}{5-2\sqrt{3}}\right)$

$I={\int }_0^{\pi /3}\frac{\cos \theta }{5-4\sin \theta }d\theta$

Let $5-4\sin \theta =t$

$\Rightarrow \cos \theta d\theta =-\frac{dt}{4}$

$I={\int }_5^{5-2\sqrt{3}}\frac{-dt}{4t}$

$I=\frac{1}{4}{\int }_{5-2\sqrt{3}}^5\frac{dt}{t}$

$I=\frac{1}{4}{\left[\log t\right]}_{5-2\sqrt{3}}^5$

$I=\frac{1}{4}[\log 5-\log (5-2\sqrt{3})]$

$I=\frac{1}{4}\left[\log \left(\frac{5}{5-2\sqrt{3}}\right)\right]$

$\therefore I=\frac{1}{4}\log \left(\frac{5}{5-2\sqrt{3}}\right)$