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

∫x*(x-2)^{-2}dx

I used integration by parts

u=x

du/dx=1

dv/dx=(x-2)^{-2}

v=-(x-2)^{-1}

Thus

∫x*(x-2)^{-2}dx=-x(x-2)^{-1}-∫1*-(x-2)^{-1}dx=-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

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# Weird integral - what's wrong?

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