Find the indefinite integral of [tex] \int\limits {\frac{5}{x^\frac{1}{2}+x^\frac{3}{2}} \, dx [/tex]I have been able to simplify it to [tex] \int\limits {\frac{5\sqrt{x}}{x^3+x}} \, dx [/tex] but that is confusing, I then did u-subsitution where [tex]u=\sqrt{x}[/tex] to obtain [tex] \int\limits {\frac{5u}{u^6+u^2}} \, dx [/tex] which simplified to [tex] \int\limits {\frac{5}{u^5+u}} \, dx [/tex], a much nicer looking integrandhowever, I am still stuckples helpshow all work or be reported

The easiest way to calculate this integral is substitution.

[tex]$\int\dfrac{5}{x^\frac{1}{2}+x^\frac{3}{2}}\,dx=5\int\dfrac{1}{x^\frac{1}{2}+(x^\frac{1}{2})^3}\,dx=5\int\dfrac{1}{\sqrt{x}+(\sqrt{x})^3}\,dx=(\star)[/tex]

Now we can substitute [tex]u=\sqrt{x}[/tex] and then:

[tex]du=\dfrac{1}{2\sqrt{x}}\,dx\qquad\implies\qquad dx=2\sqrt{x}\,du=2u\,du[/tex]

So:

[tex]$(\star)=5\int\dfrac{1}{\sqrt{x}+(\sqrt{x})^3}\,dx=5\int\dfrac{2u}{u+u^3}\,du=10\int\dfrac{1}{1+u^2}\,dx=[/tex]

[tex]=10\arctan(u)+C=\boxed{10\arctan(\sqrt{x})+C}[/tex]