Question

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

Answer 1

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

Put $x^3+1=t\Rightarrow 3x^{2}dx=dt$
i.e.,

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

$=\frac{1}{3}[\log |t-1|-\log \left|t\right|]+C$

$I=\frac{1}{3}\log \left|\frac{x^3}{x^3+1}\right|+C$