Question

how do you integrate sqrt x/(1+x^3)

Answer 1

Using chain rule we can see that: \[\sqrt{x} dx = \frac{d(\frac{2}{3}x^{\frac{3}{2}})}{dx} dx\]

\[\int\limits_{}^{}\frac{\sqrt{x}}{(1+x^3)}dx = \int\limits_{}^{}\frac{1}{(1+x^3)}d(\frac{2}{3}x^{\frac{3}{2}})= \int\limits_{}^{}\frac{1}{(1+\frac{9}{4}(\frac{2}{3}x^{\frac{3}{2}})^2)}d(\frac{2}{3}x^{\frac{3}{2}})\]

Using the integration identity \[\int\frac{1}{x^2+a^2}dx=\frac{1}{a}tan^{-1}(\frac{x}{a})\]

Using some algebraic manipulation/substitution, you should obtain:
\[\frac{2}{3}tan^{-1}(x^{\frac{3}{2}})\]