1. Nov 19, 2011

Idoubt

1. The problem statement, all variables and given/known data

Need to do this integral as part of a problem

$\int$ $\frac{xdx}{\sqrt{x^2+a^2-\sqrt{2}ax}}$

I can't think of any substitution that would work, or any way to factorize.
I think this is a standard integral, but can't remember everyone so I just want to learn to solve it.

2. Nov 19, 2011

Dick

Your first step would be completing the square in the denominator. What does the integral look like after you do that?

3. Nov 19, 2011

LCKurtz

I would try writing $x^2+a^2-\sqrt 2 ax = x^2- \sqrt 2 ax+ \frac{a^2} 2 +(a^2-\frac {a^2} 2) =(x-\frac a {\sqrt2})^2+\frac{a^2} 2$ and see if that leads anywhere.

 I see Dick types faster than I do...
 Fixed a typo and its ramifications.

Last edited: Nov 19, 2011
4. Nov 19, 2011

Dick

No, we type about the same. I just say less.

5. Nov 19, 2011

SammyS

Staff Emeritus
It appears that you have changed $\sqrt{2}ax$ into $2\sqrt{a}$, but apparently meant $2(\sqrt{a})x\,.$

6. Nov 19, 2011

LCKurtz

Yes, thanks. I edited it (about 3 times ) to correct it.

7. Nov 19, 2011

SteamKing

Staff Emeritus
You are letting the radicals and the denominator distract you. If you re-write as
Int (mess)^(-1/2) * xdx, you should see a u-substitution and integration by parts.

8. Nov 23, 2011

Idoubt

ok when I complete the squares and do it, i get the answer as

$\sqrt{x^2+a^2+\sqrt{2}ax}$ + $\frac{a}{\sqrt{2}}$ ln |$\sqrt{a^2+x^2}$ + x |

Which is the right answer i think. thanks a lot for the help :)

9. Nov 23, 2011

Idoubt

If i do the U substitution, I don't see how i can write x in terms of u