Question

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

Answer 1

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

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

Assume, $x^5=t$

Differentiate t with respect to x,

$5x^{4}dx=dt$

$=\frac{1}{5}\int \frac{dt}{t(t+3)}$

$=\frac{1}{5}.\frac{1}{3}\int (\frac{1}{t}-\frac{1}{t+3})dt$

$=\frac{1}{15}logt-\log (t+3)+c$

$=\frac{1}{15}log\left(\frac{t}{t+3}\right)+c$

$=\frac{1}{15}log\left(\frac{x^5}{x^5+3}\right)+c$