# Integrate x^3/2 divided by expression - using partial fractions perhaps

1. Apr 20, 2012

### Mustaq

1. The problem statement, all variables and given/known data
Hi. My first post!
I'm trying to solve for where a is a constant:

∫ (x/a)1/2*(x/(x-a)) dx

2. Relevant equations
See above

3. The attempt at a solution
I've tried integration by parts by setting u=(x/a)1/2 but I end up having to solve ∫ (x/a)1/2ln(x-a) - which I can't solve.
I've tried switching them around u=x/(x-a) and end up having to solve ∫x3/2/(x-a)2 - which I cant solve either.
I've thought of using partial fractions but run into x3/2/(x-a)2 which I can't do either.

Thanks

2. Apr 20, 2012

### sunjin09

The change of variable u=(x/a)1/2 is very much solvable. Try it again.

3. Apr 21, 2012

### Mustaq

So obvious. I was looking for something really complicated.
Let u = (x/a)1/2
So x=au2, dx/du=2au

First simplify:
∫(x/a)1/2(x/(x-a)) dx
∫(x/a)1/2( 1 + a/(x-a) ) dx

Substitute by u:
∫u( 1 + a/(au2-a) ) 2au du
2a∫u2( 1 + 1/(u2-1) ) du
2/3au3 + 2a∫u2/(u2-1) du
2/3au3 + 2a∫(1 + 1/(u2-1)) du
2/3au3 + 2au + aln( (u-1)/(u+1) )

Finally put back x and get required anaswer:
2/3(x3/a)1/2 + 2(ax)1/2 + aln( ((x/a)1/2-1)/((x/a)1/2+1) )

Thanks, for the hint which lead to the solution (was on it for weeks LOL)

Last edited: Apr 21, 2012