Homework Help: Integration of a natural log and polynomial

1. Sep 15, 2010

waealu

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

Evaluate the integral when x > 0:

indefinite integral of ln(x2+19x+84)dx

2. Relevant equations

I know I need to use some form of integration by parts: integral of u*dv=uv-(integral of(du*v))

3. The attempt at a solution

I began by making u=ln(x2+19x+84) and dv=dx. Thus, (after u-substitution) du=(2x+19)/(x2+19x+84) and v=x.

After putting that in the formula, we get x*ln(x2+19x+84)-(integral of)((2x2+19x)/(x2+19x+84)). After simplifying that, I get:

x*ln(x2+19x+84)-((x2+19x+84)(4x+19)-(2x2+19x)(2x+19))/((x2+19x+84)2)

But according to the program I am using, that is the incorrect answer. Do you have any suggestions? Thanks.

Last edited: Sep 15, 2010
2. Sep 15, 2010

jackmell

Why not just factor the quadratic, then split up the integral into two simpler log terms then use:

$$\int ln(u)du=u\ln(u)-u$$

3. Sep 15, 2010

losiu99

Hmm, by infinite do you mean definite integral from 0 to infinity? If so, it's clearly divergent.

4. Sep 15, 2010

waealu

No, sorry, I meant the indefinite integral.

5. Sep 15, 2010

waealu

Thanks Jackmell. I tried that method and it worked. (A lot easier than the method I was using.)