Mathematics

Evaluate $$\int \cot^{2}x\ dx$$.


SOLUTION
$$\int{\cot ^{2}x}\space dx$$

We know that  $$\text{cosec}^2x-\cot ^2x=1$$
Therefore, $$\Rightarrow \cot ^2x=\text{cosec}^2x-1$$

$$\Rightarrow \int{\cot ^{2}x}\space dx=\int{(\text{cosec}^2x-1)dx}$$

$$\Rightarrow \int{\text{cosec}^2x\space dx}-\int{1}\space dx$$

This can be written as 
$$\Rightarrow -\int{-\text{cosec}^2x\space dx}-\int{1}\space dx$$

This is a direct integral 

$$\Rightarrow -(\cot x)-x+C$$

$$\int{\cot ^{2}x}\space dx=-\cot x-x+C$$
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Subjective Medium Published on 17th 09, 2020
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