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## Main Question or Discussion Point

Heya...

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

∫x*(x-2)

I used integration by parts

u=x

du/dx=1

dv/dx=(x-2)

v=-(x-2)

Thus

∫x*(x-2)

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*(x-2)

as a result...

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

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

∫x*(x-2)

^{-2}dxI 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)+CNow... 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)+Cas a result...

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