The Solution:
[tex]\int(x+3)\sqrt{5-x}\text{ }dx[/tex]
We are required to find the indefinite integral.
Step 1:
[tex]\begin{gathered} Let\text{ }u=5-u \\ \frac{du}{dx}=-1 \\ dx=-du \end{gathered}[/tex][tex]\begin{gathered} x=5-u \\ \\ x+3=5-u+3=8-u \end{gathered}[/tex]
Substituting, we get
[tex]\int-(8-u)\sqrt{u}\text{ }du=\int(u-8)u^{\frac{1}{2}}\text{ }du=\int u^{\frac{3}{2}}-8u^{\frac{1}{2}}\text{ }du[/tex][tex]\begin{gathered} =\frac{2}{5}u^{\frac{5}{2}}-\frac{16}{3}u^{\frac{3}{2}}+C \\ \\ \text{ Substituting }5-x\text{ for u, we get} \\ \\ =\frac{2}{5}(5-x)^{\frac{5}{2}}-\frac{16}{3}(5-x)^{\frac{3}{2}}+C \end{gathered}[/tex]
Therefore, the corect answer is
