Question

Evaluate:
$\int \frac{1}{x^2(x^4+1{)}^{3/4}}dx$

Answer 1

$I=\int \frac{1}{x^2.c^3(1+x^{-4}{)}^{3/4}}dx=\int \frac{x^{-5}}{(1+x^{-4}{)}^{3/4}}dx$

Let $1+x^{-4}=t^4$

$\Rightarrow -4x^{-5}dx=4t^{3}dt$

$\Rightarrow I=\frac{-1}{4}\int \frac{dt\times 4t^3}{t^3}$

$=-t+c=-(1t+x^{-4})+c$

$=-1-\frac{1}{x^4}+c$.