x^7 (1+x^4 )^2 d x - Vidyadip

Question

$\int \frac{x^7}{\left(1+x^4\right)^2} d x$

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Answer

$ \text { Let } I =\int \frac{x^7}{\left(1+x^4\right)^2} d x$
$=\int \frac{x^4 \cdot x^3}{\left(1+x^4\right)^2} d x $
Put $1+x^4=t$
$ \therefore 4 x ^3 dx = dt$
$\therefore x ^3 d x =\frac{1}{4} dt$
$\therefore I =\frac{1}{4} \int \frac{( t -1)}{ t ^2} dt$
$=\frac{1}{4}\left(\int \frac{1}{ t } dt -\int \frac{1}{ t ^2} dt \right)$
$=\frac{1}{4}\left[\log | t |-\frac{ t ^{-1}}{-1}\right]+ c$
$\therefore \frac{1}{4}\left[\log \left|1+x^4\right|+\frac{1}{1+x^4}\right]+ c $

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