Partial fraction integral I didn't know how to do

Question

I had my 2nd Calc II test today and I felt I did horrible. One problem that I could figure out was a partial fractions.

THe integral 4/(x^2+6x+34)

I completed the square in denominator(x^2+3)^2 but I was left with a +25 that I didn't know what to do with. I ended up setting my equation up like this

Ax+B/(x^2+3) + Bx+C(x^2+3)^2

Answer 1

It's already in partial fraction form, unless you want to involve complex numbers. Completing the square was a good idea you get $$\int \frac{4}{(x+3)^2 + 5^2} \, \mathrm{d}x = \int \frac{4}{u^2 + 5^2} \, \mathrm{d}u$$ using $u = x+3$. And then either use $u = 5\tan x$ or note that $\int \frac{\mathrm{d}x}{x^2 + a^2} = \frac{1}{a}\arctan \frac{x}{a} + \mathrm{c}$ (which the substitution shows)