Integrate x/(x^2-2x+5): Clarifying Steps

[Azureflames]

Homework Statement

Hi, I'm trying to take the integral of x/(x^2-2x+5) dx but I'm not sure what to do.

Homework Equations

The Attempt at a Solution

I started by completeing the square in the denominator giving me the integral of x/((x-1)^2+4)) dx but I am not sure where to go from there. I have the correct answer for it but I need to understand the steps involved getting there.

EDIT: Okay, after spending a decent amount of time on this problem I finally look for some place to get help, and 5 minutes later I think I come up with a solution.

First, I set u = x^2 - 2x +5, du/2 - 2/2 = x. Substituted back in which gave me (1/2)integral( (du-2)/u ) which I then split into (1/2)integral(du/u) - (1/2)integral(2/u).

Taking the integral of the first part gave me (1/2)ln(x^2 - 2x +5). For the second half, I substitued the u values back into the equation which gave me: -(1/2)integral(2/(x^2 - 2x + 5). I completed the square in the denominator which gave me -integral( 1/((x-1)^2+4 ).

Integrating that part of the equation gives me -(1/2)arctan((x-1)/2).

So my final answer is (1/2)ln(x^2 - 2x +5) - (1/2)arctan((x-1)/2) + C. Can someone please confirm my steps? Sorry if my work is hard to follow. I wasn't sure how to make the proper symbols and my time is short :)

 

[Dick]

That's right.

 

[blochwave]

Hell, that seemed clever(or I suck)

I would've done partial fraction decomposition, and heaven knows NO ONE likes doing that