What's Wrong with My Integration by Parts?

[QuickQThrowaway]

Heya...

I just stumbled upon this integral while doing ODEs...

∫x*(x-2)^-2dx

I used integration by parts

u=x
du/dx=1
dv/dx=(x-2)^-2
v=-(x-2)^-1

Thus

∫x*(x-2)^-2dx=-x(x-2)^-1-∫1*-(x-2)^-1dx=-x*(x-2)^-1+ln(x-2)+C

Now... I don't see anything wrong with this...

But every single online integration calculator as well as the ODE solution solves this integral by doing 1/(x-2)+2/(x-2)² and they all get

-2*(x-2)^-1+ln(x-2)+C

as a result...

What am I doing wrong in my integration by parts? What am I missing? Oo

 

[mathman]

There is nothing wrong. The two results are equivalent.

$\frac{x}{x-2}=\frac{x-2+2}{x-2}=1+\frac{2}{x-2}$.

The difference is only in the constant of integration.

 

[QuickQThrowaway]

There is nothing wrong. The two results are equivalent.

$\frac{x}{x-2}=\frac{x-2+2}{x-2}=1+\frac{2}{x-2}$.

The difference is only in the constant of integration.

Thank you!
The maths department screwed up then because the question asked to verify that C=2 but with the above integration it equals 3... So they omitted one result... >.<