MHB What is the integral of $\frac{x}{2\sqrt{x+2}}$?

[karush]

$\tiny\text{LCC 206 {r9} IBP}$
$I=\int \frac{x}{2\sqrt{x+2}}\,dx
=\frac{\left(x-4\right)\sqrt{x+3}}{3}$
Rewrite as
$I=\int \frac{x}{2}\frac{1}{\sqrt{x+2}}\, dx $
Let
$\displaystyle
u=\frac{x}{2} \ \ \ dv=\frac{1}{\sqrt{x+2}} \, dx \\
du=\frac{1}{2} \ \ \ v=2\sqrt{x+2} \\ $
Then
$x\sqrt{x+2}-\int \sqrt{x+2} \ dx$
This doesn't look it's heading toward the answer

$\tiny\text
{from Surf the Nations math study group}$

 

[MarkFL]

You're doing fine...complete the last integral (with a constant of integration), then factor. :)

 

[karush]

$\tiny\text{LCC 206 {r9} IBP}$
$$\displaystyle
I=\int \frac{x}{2\sqrt{x+2}}\,dx
=\frac{\left(x-4\right)\sqrt{x+3}}{3} \\
\text{rewrite as} \\
I=\int \frac{x}{2}\frac{1}{\sqrt{x+2}}\, dx \\
u=\frac{x}{2} \ \ \ dv=\frac{1}{\sqrt{x+2}} \, dx \\
du=\frac{1}{2} \ \ \ v=2\sqrt{x+2} \\
\text{then} \\
I=x\sqrt{x+2}-\int \sqrt{x+2} \ dx \\
= x\sqrt{x+2}-\frac{2}{3}\left(x+2\right) \\
\text{factor } \\
=\sqrt{x+2}\left[x-\frac{2\left(x+2\right)}{3}\right] \\
\text{simplify} \\
I=\frac{\left(x-4\right)\sqrt{x+3}}{3}$$
$\tiny\text
{from Surf the Nations math study group}$