$\displaystyle \int x^4\sqrt{4+x^5}$
Do I make $\displaystyle u=4+x^5$ in order to solve this problem?
Yes.
Notice that if $\displaystyle u = 4 + x^5$ then $\displaystyle \frac{du}{dx} = 5x^4$.
So $\displaystyle \int{x^4\sqrt{4 + x^5}\,dx} = \frac{1}{5}\int{5x^4(4 + x^5)^{\frac{1}{2}}\,dx}$
$\displaystyle = \frac{1}{5}\int{u^{\frac{1}{2}}\,\frac{du}{dx}\,dx }$
$\displaystyle = \frac{1}{5}\int{u^{\frac{1}{2}}\,du}$.
I trust you can go from here...
Integration by substitution is like a backward-application of the chain rule. In the expression
$\displaystyle x^4\sqrt{4+x^5},$
we see that $\displaystyle x^4$ looks like it was differentiated from $\displaystyle 4+x^5$ of the square root.
To find the real antiderivative, we just add in the factor $\displaystyle \frac{1}{5}$ to get $\displaystyle x^4$ from $\displaystyle 5x^4$.