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
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