Consider the integral $$\int\frac{x}{\sqrt{x^2+8 x}}dx$$ Then an appropriate trigonometric substitution to simplify this integral is $x=f(t)$ where
$f(t)$ =
After making this substitution and simplifying, we obtain the integral $\displaystyle \int g(t)\,dt$ where
$g(t)$ =

Note that this problem doesn't ask you to evaluate this integral.