Question

$\int \frac{dx}{x(x^5+3)}$

Answer 1

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

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

Let $t=x^5$

$5x^{4}dx=dt$

Therefore,

$I=\int \frac{dt}{5t(t+3)}$

$I=\frac{1}{5}\int \frac{1}{3}[\frac{1}{t}-\frac{1}{t+3}]dt$

$I=\frac{1}{15}\log |\frac{t}{t+3}|+C$

$I=\frac{1}{15}\log |\frac{x^5}{x^5+3}|+C$