MHB Integration by Parts with Domain Warning

[karush]

$\int x\ \cot^2\left({x}\right) dx $
$u=x$ $dv=\cot^2\left({x}\right) dx $
$du=\frac{x^2}{2}$ $v=\frac{-\cos\left({x}\right)+x\sin\left({x}\right)}{\sin\left({x}\right)}$

 

[Dethrone]

Hi karush,

Could you show your workings for $v$?
We have $dv=\cot^2\left({x}\right) dx$
$v=\int (\csc^2 x -1) \,dx$

Can you proceed?

 

[MarkFL]

I agree with you choice of $u$ and $dv$:

$$u=x\,\therefore\,du=dx$$ (You integrated rather than differentiated)

$$dv=\cot^2(x)\,dx=\left(\csc^2(x)-1\right)\,dx\,\therefore\,v=?$$

 

[karush]

That what I got for $v$ with the TI ??

 

[MarkFL]

That what I got for $v$ with the TI ??

I suspect your TI spat out:

$$v=-\frac{\cos\left({x}\right)+x\sin\left({x}\right)}{\sin\left({x}\right)}$$ :D

 

[karush]

$uv-\int\ v\ du$

$-x(\cot\left({x}\right)-x)+\int \cot\left({x}\right)dx -\int x\ dx$

My TI returned a domain warning??