1. Tardigrade
  2. Question
  3. Mathematics
  4. The value of ∫ limits(π/3)(π/2) ((2+3 sin x)/ sin x(1+ cos x)) d x is equal to

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. The value of $\int\limits_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{(2+3 \sin x)}{\sin x(1+\cos x)} d x$ is equal to

JEE MainJEE Main 2023Integrals

Solution:

$ \int\limits_{\pi / 3}^{\pi / 2}\left(\frac{2+3 \sin x}{\sin x(1+\cos x)}\right) d x=2 \int\limits_{\pi / 3}^{\pi / 2} \frac{d x}{\sin x+\sin x \cos x}+3$
$ 3 \int\limits_{\pi / 3}^{\pi / 2} \frac{d x}{1+\cos x} $
$\int\limits_{\pi / 3}^{\pi / 2} \frac{d x}{1+\cos x}=\int\limits_{\pi / 3}^{\pi / 2} \frac{1-\cos x}{\sin ^2 x} d x $
$ =\int\limits_{\pi / 3}^{\pi / 2}\left(\text{cosec}^2 x-\cot x \text{cosec} x\right) d x$
$ =(\operatorname{cosec} x-\cot x) \int\limits_{\pi / 3}^{\pi / 2}=(1)-\left(\frac{2}{\sqrt{3}}-\frac{1}{\sqrt{3}}\right)=1-\frac{1}{\sqrt{3}} $
$ \int\limits_{\pi / 3}^{\pi / 2} \frac{d x}{\sin x(1+\cos x)}= $
$ \int \frac{d x}{(2 \tan x / 2)\left(1+1-\tan ^2 x / 2\right)}$
$ =\int \frac{\left(1+\tan ^2 x / 2\right) \sec ^2 x / 2 d x}{2 \tan x / 2}$
$ \tan x / 2= t $
$ \sec x / 2 \frac{1}{2} dx = dt$
$ \frac{1}{2} \int\left(\frac{1+ t ^2}{ t }\right) dt =\frac{1}{2}\left[\ln +\frac{ t }{2}^2\right]_{\frac{1}{\sqrt{3}}}^1 $
$ =\frac{1}{2}\left[\left(0+\frac{1}{2}\right)-\left(\ln \frac{1}{\sqrt{3}}+\frac{1}{6}\right)\right]=\left(\frac{1}{3}+\ln \sqrt{3}\right) \frac{1}{2} $
$ =\left(\frac{1}{6}+\frac{1}{2} \ln \sqrt{3}\right) $
$ 2\left(\frac{1}{6}+\frac{1}{2} \ln \sqrt{3}\right)+3\left(1-\frac{1}{\sqrt{3}}\right) $
$ =\frac{1}{3}+\ln \sqrt{3}+3-\sqrt{3}=\frac{10}{3}+\ln \sqrt{3}-\sqrt{3}$