Question

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

Answer 1

Let $I=\int \frac{1}{x^2{(x^4+\frac{1}{1})}^{3/4}}$

$I=\int \frac{1}{x^5{(1+\frac{1}{x^4})}^3}dx$

Let $1+\frac{1}{x^4}=Z$

So, $\frac{-4}{x^5}dx=dZ$

$\frac{1}{x^5}=\frac{-1}{4}dZ$

So, $I=\int \frac{dZ}{-4Z^3}=\frac{1}{8Z^2}+C$

$I=\frac{1}{8{(1+\frac{1}{x^4})}^2}+C=\frac{x^8}{8{(1+x^4)}^2}+C$