integration by parts with my work

mpgcbball

1. The problem statement, all variables and given/known data

integrate arctan(1/x)

2. Relevant equations



3. The attempt at a solution

z=arctan(1/x)
dx=-dz(x^2-1)

now its the integral of z(x^2-1)dz

let u =X^2-1
du=2x
dv=-udu
v=-u^2/2

integral=(x^2-1)(-u^2/2) - int (-u^2)(2x)

this is where i got stuck but i think im doing the z substitution incorrectly. is it even necessary to sub z?

Thanks!

cristo

You got stuck because you're trying to integrate the term z(x^2-1)dz which has x's and z's in it!

We want to calculate [tex]\int\tan^{-1}(1/x)dx[/tex]. Do this by parts, and take u=arctan(1/x) and dv=dx. You need to then calculate du and v, and use the usual integration by parts forumla: [tex]\int udv= uv-\int vdu[/tex]

(Alternatively, you could note that arctan(1/x)=arccot(x) and proceed from here)