Integrating a function that's proportional to...

I am supposed to integrate a function that is proportional to

$\displaystyle (10+x)^{-2}$

My initial strategy was to treat that as the reciprocal of

$\displaystyle x^2 + 20x + 100$

but that leaves me with the whole thing raised to an exponent of -1. I don't know how to take an antiderivative anything raised to -1!

So I decided to make a new variable, A, and set that equal to x+10. Taking the antiderivative of

$\displaystyle A^{-2}$

was pretty easy, giving me

$\displaystyle -1/A = -(x+10)^{-1}$

I just don't know if what I did was mathematically correct. (Getting a definite integral at that point was easy, and resulted in an answer close to one of my multiple choices.)

The thing is, I'm treating the whole thing as if I were getting the antid. of

$\displaystyle (10+x)^{-2}$.

Really I'm just dealing with an unknown function proportional to that. Is this something I haven't learned yet

(Worried)

or can I just take the antiderivative of what my function is proportional to?