# Integration problem. Do they want me to use partial fractions?

• Oct 20th 2010, 03:53 AM
bjorno
Integration problem. Do they want me to use partial fractions?
I am working on an assignment. The first task is:

Find the indefinite integral of (x+2)/(x^2)+x

I factored out an x of the denominator, and split to partial fractions that were easy to integrate. My result was: 2*ln|x| - ln|x+1|

My questions are:
Have i done it right?
Even if i have, is there another (simpler, or more clever) way to do this?
• Oct 20th 2010, 04:30 AM
Prove It
I assume you are actually trying to write

$\displaystyle{\int{\frac{x+2}{x^2+x}\,dx}}$

in which case, partial fractions is the way to go.
• Oct 20th 2010, 05:04 AM
pirateboy
If by partial fractions you mean partial fraction decomposition, you don't even have to do that. You just have to see that
$\displaystyle \frac{x+2}{x^2 + 2} = \frac{x}{x^2+2}+\frac{2}{x^2+2}$

You'll then end up with two separate integrals you'll have to bust some sweet moves on. One should be straight up u-substitution and the other well...here's a BIG hint (it's not a bad idea to have this memorized)

$\displaystyle \int\frac{dx}{a^2+x^2} = \frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right) + C$
• Oct 20th 2010, 05:05 AM
bjorno
Great, thanks!
• Oct 20th 2010, 06:49 AM
Prove It
Quote:

Originally Posted by pirateboy
If by partial fractions you mean partial fraction decomposition, you don't even have to do that. You just have to see that
$\displaystyle \frac{x+2}{x^2 + 2} = \frac{x}{x^2+2}+\frac{2}{x^2+2}$

You'll then end up with two separate integrals you'll have to bust some sweet moves on. One should be straight up u-substitution and the other well...here's a BIG hint (it's not a bad idea to have this memorized)

$\displaystyle \int\frac{dx}{a^2+x^2} = \frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right) + C$

Except it's not $\displaystyle{\frac{x + 2}{x^2 + 2}}$, it's $\displaystyle{\frac{x+2}{x^2 + x}}$.