Mathematics

Evaluate the given integral$\int { x.cosec ^{ 2 }{ x } } dx$

SOLUTION
$I=\displaystyle\int{x.{cosec}^{2}{x}dx}$

Integrating by parts,we get

Let $u=x\Rightarrow\,du=dx$

$dv={cosec}^{2}{x}dx\Rightarrow\,v=-\cot{x}$

$\int u.v dx=u \int vdx-\int \left [\int vdx. \dfrac{du}{dx}.dx \right ]$......by parts formula.

$I=-x\cot{x}-\displaystyle\int{-\cot{x}dx}$

$I=-x\cot{x}+\displaystyle\int{\cot{x}dx}$

We know that $\displaystyle\int{\cot{x}dx}=\log{\left|\sin{x}\right|}+c$

$\therefore\,I=-x\cot{x}+\log{\left|\sin{x}\right|}+c$

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

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