The value of 3 / 2 ( 2 + 3 sin x ) sin x ( 1 + cos x ) d x is equal to [2023]

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Solution

Q.
The value of $\int_{π / 3}^{π / 2} \frac{(2 + 3 \sin x)}{\sin x (1 + \cos x)} d x$ is equal to [2023]

1 $\frac{7}{2} - \sqrt{3} - \log_{e} \sqrt{3}$

2 $\frac{10}{3} - \sqrt{3} - \log_{e} \sqrt{3}$

3 $\frac{10}{3} - \sqrt{3} + \log_{e} \sqrt{3}$

4 $- 2 + 3 \sqrt{3} + \log_{e} \sqrt{3}$

Ans.
(3)

Let $I = \int_{π / 3}^{π / 2} \frac{2 + 3 \sin x}{\sin x (1 + \cos x)} d x$

$= \int_{π / 3}^{π / 2} \frac{2 (1 - \cos x)}{\sin x (1 - \cos^{2} x)} d x + \int_{π / 3}^{π / 2} \frac{3}{(1 + \cos x)} d x$

$= \int_{π / 3}^{π / 2} \frac{2}{\sin^{3} x} d x - \int_{π / 3}^{π / 2} \frac{2 \cos x}{\sin^{3} x} d x + \frac{3}{2} \int_{π / 3}^{π / 2} \frac{1}{\cos^{2} \frac{x}{2}} d x$

$= 2 \int_{π / 3}^{π / 2} \sqrt{c o s e c^{2} x} c o s e c^{2} x d x - 2 \int_{π / 3}^{π / 2} \cot x \cdot c o s e c^{2} x d x + \frac{3}{2} {[2 \tan \frac{x}{2}]}_{π / 3}^{π / 2}$

$= 2 \int_{π / 3}^{π / 2} \sqrt{1 + \cot^{2} x} c o s e c^{2} d x - 2 \int_{π / 3}^{π / 2} \cot x \cdot c o s e c^{2} d x + 3 [\tan \frac{π}{4} - \tan \frac{π}{6}]$

Let $\cot x = t \Rightarrow c o s e c^{2} x d x = - d t$ and

For $x = \frac{π}{2}; t = 0$ and for $x = \frac{π}{3}; t = \frac{1}{\sqrt{3}}$

$∴$ $I = - 2 \int_{1 / \sqrt{3}}^{0} \sqrt{1 + t^{2}} d t + 2 \int_{1 / \sqrt{3}}^{0} t d t + 3 [1 - \frac{1}{\sqrt{3}}]$

$= - 2 {[\frac{t}{2} \sqrt{1 + t^{2}} + \frac{1}{2} \log_{e} (t + \sqrt{1 + t^{2}})]}_{1 / \sqrt{3}}^{0} + {[t^{2}]}_{1 / \sqrt{3}}^{0} + (3 - \sqrt{3})$

$= \frac{2}{3} + \log \sqrt{3} - \frac{1}{3} + 3 - \sqrt{3} = \frac{10}{3} - \sqrt{3} + \log_{e} \sqrt{3}$

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