Question

Evaluate: $\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{cos\theta }{\left(1+sin\theta \right)\left(2+sin\theta \right)}d\theta$

• A. $log\left(\frac{4}{3}\right)$
• B. $log\left(\frac{3}{4}\right)$
• C. $log4+log3$
• D. None of these

Answer 1

Answer: (A)

Let $sin\theta =t\Rightarrow cos\theta d\theta =dt$

when $\theta =0,t=sin0=0$ {\,}and when $\theta =\frac{\pi }{2},sin\frac{\pi }{2}=1$

$\therefore \underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{cos\theta }{\left(1+sin\theta \right)\left(2+sin\theta \right)}d\theta$

$=\underset{0}{\overset{1}{\int }}\frac{1}{\left(1+t\right)\left(2+t\right)}dt=\underset{0}{\overset{1}{\int }}\left(\frac{1}{1+t}-\frac{1}{2+t}\right)dt$ [By using partial fractions]

$={\left[log\left(1+t\right)\right]}_0^1-{\left[log\left(2+t\right)\right]}_0^1$

$=log2-log3+log2=2log2-log3=log\left(\frac{4}{3}\right)$