Answer
$$\int\frac{1}{x\sqrt{x+9}}dx=\frac{1}{3}\ln|\frac{\sqrt{x+9}-3}{\sqrt{x+9}+3}|+C$$
Work Step by Step
$$I=\int\frac{1}{x\sqrt{x+9}}dx$$
We set $u^2=x+9$, so $$2udu=dx$$
Also, $x=u^2-9$ and $\sqrt{x+9}=u$
$$I=\int\frac{2udu}{(u^2-9)u}=\int\frac{2}{u^2-9}du$$ $$I=\int\frac{2}{(u-3)(u+3)}du$$ $$I=\frac{1}{3}\int\frac{6}{(u-3)(u+3)}du$$ $$I=\frac{1}{3}\int\frac{(u+3)-(u-3)}{(u-3)(u+3)}du$$ $$I=\frac{1}{3}\Big(\int\frac{du}{u-3}-\int\frac{du}{u+3}\Big)$$ $$I=\frac{1}{3}(\ln|u-3|-\ln|u+3|)+C$$ $$I=\frac{1}{3}\ln|\frac{u-3}{u+3}|+C$$ $$I=\frac{1}{3}\ln|\frac{\sqrt{x+9}-3}{\sqrt{x+9}+3}|+C$$
