Question

Find the indefinite integral by making a change of variables. (Use C for the constant of integration.)

Answer 1

The Solution:

`\int(x+3)√(5-x)\text{ }dx`

We are required to find the indefinite integral.

Step 1:

`\begin{gathered} Let\text{ }u=5-u \\ (du)/(dx)=-1 \\ dx=-du \end{gathered}`
`\begin{gathered} x=5-u \\ \\ x+3=5-u+3=8-u \end{gathered}`

Substituting, we get

`\int-(8-u)√(u)\text{ }du=\int(u-8)u^{(1)/(2)}\text{ }du=\int u^{(3)/(2)}-8u^{(1)/(2)}\text{ }du`
`\begin{gathered} =(2)/(5)u^{(5)/(2)}-(16)/(3)u^{(3)/(2)}+C \\ \\ \text{ Substituting }5-x\text{ for u, we get} \\ \\ =(2)/(5)(5-x)^{(5)/(2)}-(16)/(3)(5-x)^{(3)/(2)}+C \end{gathered}`

Therefore, the corect answer is