Why Does Integrating (x+2)/(x^2+4x) Lead to a Logarithmic Function?

[vipertongn]

S (x+2)/(x^2+4x) dx

I've been learning about natural logs and their properties but at this answer I get befuddled at how to work it out. I think perhaps substitution, and some properties with logs but I am very weak in logs...Didn't do well in it when I was in algebra.
I want to know how to do this equation, although I know it has this as an answer:

log(abs(x^2+4x))/2

Thanks for any help.

 

[quantumdude]

I think perhaps substitution

You think rightly. Now take a stab at choosing the right substitution. There aren't that many choices available, so this shouldn't be too hard.

 

[vipertongn]

hmmm ok let's see u = x^2+4x --> du=2x+4 --> du/2=x+2

makes the equation

1/2 S du/U but 1/u means ln|u| though right? why is it a log?

 

[Mark44]

hmmm ok let's see u = x^2+4x --> du=2x+4 --> du/2=x+2

makes the equation

1/2 S du/U but 1/u means ln|u| though right? why is it a log?

That's the right substitution, but you're missing something in du.
u = x^2 + 4x ==> du = (2x + 4)dx

$$\int du/u = ln |u| + C$$
An antiderivative of 1/u is ln |u | because the derivative of ln |u| is 1/u. It's a sort of inverse relationship, similar to the relationship between the equations y = ln(x) and x = e^y.