Question

Integrate

$\int \frac{x^5}{\sqrt{1+x^3}}dx$

Answer 1

$\int \frac{x^5}{\sqrt{1+x^3}}dx$

Let $1+x^3=t^2$

$⟹3x^{2}dx=2tdt$

$⟹x^{2}dx=\frac{2}{3}tdt$

$=x^{2}dx=\frac{2}{3}tdt$

$\int \frac{\frac{2}{3}t(t^2-1)}{t}dt$

$=\frac{2}{3}\int (t^2-1)tdt$

$=\frac{2}{3}$ ( $\frac{{(1+x^3)}^{3/2}}{3}-(1+x^3{)}^{1/2})$