Where is my mistake in integrating x/(x+√(x+2)) with v-sub?

[Ayesh]

Homework Statement

$$\int \frac {x}{x+ \sqrt {x+2}}$$

The Attempt at a Solution

I used the v-sub.

I also did a long division.

It gave me $$\int 2v-2- \frac {v}{v-1} + \frac {2}{v-1}$$

My final answer $$2x+2-2 \sqrt {x+2} -ln \sqrt {x+2} -1 +2ln \sqrt {x+2} -1 +c$$

The final answer should be $$x-2 \sqrt {x+2} - 2/3ln \sqrt {x+2} -1 + \frac 8/3ln \sqrt {x+2} + 2 +c$$

Where is my mistake?

 

[vela]

Homework Statement

$$\int \frac {x}{x+ \sqrt {x+2}}$$

The Attempt at a Solution

I used the v-sub.

I also did a long division.

It gave me $$\int 2v-2- \frac {v}{v-1} + \frac {2}{v-1}$$

My final answer $$2x+2-2 \sqrt {x+2} -ln \sqrt {x+2} -1 +2ln \sqrt {x+2} -1 +c$$

The final answer should be $$x-2 \sqrt {x+2} - 2/3ln \sqrt {x+2} -1 + \frac 8/3ln \sqrt {x+2} + 2 +c$$

Where is my mistake?

Kind of hard to say when you leave out critical details, like what substitution you actually used.

 

[Ayesh]

I attached a file with what I have done, since I am very bad with latex code.

 

[Mark44]

You still didn't tell us what the substitution was that you used. If you want help, you need to provide the necessary information.

 

[vela]

Looks like your polynomial division is wrong. The remainder should be over v^2+v-2, and there appears to be a sign error as well. The quotient looks okay, though.