Mathematics

# $\int { { cosec }^{ 3 }x } dx$

##### SOLUTION
$\int \csc ^3xdx$

substitute $u=\tan \left ( \dfrac{x}{2} \right )$

$\therefore \int \csc ^3xdx=\int \dfrac{(1+u^2)^2}{4u^3}du$

$\int \dfrac{(1+u^2)^2}{4u^3}du = \dfrac{1}{4}\int\left ( \dfrac{1}{u^3}+\dfrac{2}{u}+u \right ) du$

$= \dfrac{1}{4}\left [ -\dfrac{1}{2u^2}+2\ln u+\dfrac{u^2}{2} \right ]$

$= \dfrac{1}{4}\left [ -\dfrac{1}{2\tan ^2\left ( \frac{x}{2} \right )}+2\ln \tan \left ( \dfrac{x}{2} \right )+\dfrac{\tan \left ( \frac{x}{2} \right )^2}{2} \right ]+C$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

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If $A^{\prime}$ s income is $30$ $\%$ less than $B^{\prime}$ s, then how much per cent is $B^{\prime}$ s income more than $A^{\prime}$ s?
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1 Verified Answer | Published on 17th 09, 2020

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