Question

Integrate :
$\int \frac{dx}{x\left(x^3+1\right)}$

Answer 1

Consider the given integral.

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

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

Let $t=1+\frac{1}{x^3}$

$\frac{dt}{dx}=0-\frac{3}{x^4}$

$-\frac{dt}{3}=\frac{dx}{x^4}$

Therefore,

$I=-\frac{1}{3}\int \frac{dt}{t}$

$I=-\frac{1}{3}\ln \left(t\right)+C$

On putting the value of $t$, we get

$I=-\frac{1}{3}\ln (1+\frac{1}{x^3})+C$

Hence, this is the answer.