Integrate the following functions w.r.t. x: (x-1)^2 (x^2+1 )^2 - Vidyadip

Question

Integrate the following functions w.r.t. x:
$\frac{(x-1)^2}{\left(x^2+1\right)^2}$

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Answer

Let $I=\int \frac{(x-1)^2}{\left(x^2+1\right)^2} d x$$=\int \frac{x^2-2 x+1}{\left(x^2+1\right)^2} d x$
$=\int \frac{\left(x^2+1\right)-2 x}{\left(x^2+1\right)^2} d x$
$=\int\left[\frac{x^2+1}{\left(x^2+1\right)^2}-\frac{2 x}{\left(x^2+1\right)^2}\right] d x$
$=\int \frac{1}{x^2+1} d x-\int \frac{2 x}{\left(x^2+1\right)^2} d x$
$=I_1-I_2$
In $I_2$, Put $x^2+1=t \quad \therefore 2 x d x=d t$
$ \therefore I =\int \frac{1}{x^2+1} d x-\int t^{-2} d t$
$ =\tan ^{-1} x-\frac{t^{-1}}{(-1)}+c$
$ =\tan ^{-1} x+\frac{1}{x^2+1}+c .$

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