Answer
$506$
Work Step by Step
Let $u=x+4$, then $du=dx$ and when $x:0\to 5$, then $u:4\to 9$. Now, we have
We have
\begin{align*}
\int_{0}^{5} 15x(x+4)^{1/2} dx&=\int_{4}^{9} 15(u-4)(u)^{1/2} du\\
&=\int_{4}^{9} 15u^{3/2}-60u^{1/2} du\\
&=\frac{15}{5/2}u^{5/2}-\frac{60}{3/2}u^{3/2}|_4^9\\
&=6(3^5-2^5)-40(3^3-2^3)\\
&=506.
\end{align*}
