Integration by Parts for ln(sqrt(x^2-4)) | Solving Tricky Integration Problems

[Bryon]

Homework Statement

The the problem is integrate ln(sqrt(x^2-4)

Homework Equations

For this one u-sub does not work, which leads me to think that integration by parts is the way to go but am not sure how to finish it.

The Attempt at a Solution

u =1/2ln(x^2-4)
du = (1/2)(2x/(x^2-4)) = x^2/(x^2-4)

dv = dx ----> int dx = x = v

x(1/2ln(x^2-4)) - int (x^2)/(X^2-4)

I don't know what to do about the part in bold. I am thinking integration by part again but that doesn't look like it will work, unless there is one more integration to do. Any idea what i can try?

Thanks!

 

[holezch]

I think parts can work again, since 1 / x^2 - 4 is easy enough to integrate, then you'll need to integrate xarctan(x/2) which should be fine

 

[Dick]

Use polynomial division to write x^2/(x^2-4)=1+4/(x^2-4). Looks more to me like 'partial fractions' than 'parts'.

 

[Gavins]

Write it as
ln(x^2 - 4)^1/2

and also note that it's a difference of 2 squares. Then just use the log properties to reduce this to something easier and you can then use integration by parts.

 

[Bryon]

Thanks Dave! I did not see that, but it certainlly makes sense now.