Question

Integrate:

$\int \frac{dx}{x^5(1+x^{-4})}$

Answer 1

Consider the given integral.

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

Let $t=x^{-4}$

$\frac{dt}{dx}=-4x^{-5}$

$-\frac{dt}{4}=\frac{dx}{x^5}$

Therefore,

$I=-\frac{1}{4}\int \frac{dt}{(1+t)}$

$I=-\frac{1}{4}\ln (1+t)+C$

On putting the value of $t$, we get

$I=-\frac{1}{4}\ln (1+x^{-4})+C$

$I=-\frac{1}{4}\ln (1+\frac{1}{x^4})+C$

$I=-\frac{1}{4}\ln \left(\frac{x^4+1}{x^4}\right)+C$

$I=\frac{1}{4}\ln \left(\frac{x^4}{x^4+1}\right)+C$

Hence, this is the answer.