Question

Evaluate $\int \frac{dx}{x(x^3+8)}$.

Answer 1

Let $I=\int \frac{dx}{x(x^3+8)}=\frac{1}{3}\int \frac{3x^{2}dx}{x^3(x^3+8)}$

Put $x^3+8=t\Rightarrow 3x^{3}dx=dt$

$I=\frac{1}{3}\int \frac{dt}{(t-8)t}\Rightarrow \frac{1}{3}\int \frac{1}{8}\times (\frac{1}{t-8}-\frac{1}{t})$

$\Rightarrow \frac{1}{24}[\log |t-8|-\log \left|t\right|]+C$

$I=\frac{1}{24}\log \left|\frac{x^3}{x^3+8}\right|+C$