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  4. Evaluate: ∫ limits0(π/2) (cos θ/(1+sin θ)(2+sin θ )) dθ

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Q. Evaluate: $\int\limits_{0}^{\frac{\pi}{2}} \frac{cos\,\theta}{\left(1+sin\,\theta\right)\left(2+sin\,\theta \right)} d\theta $

Integrals

Solution:

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

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

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

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

$ = \left[log\left(1+t\right) \right]_{0}^{1}-\left[log\left(2+t\right)\right]_{0}^{1}$

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