Find the integral of 2x/(x^2 + 6x + 13)dx

Answer 1

∫ 2xdx/(x^2 + 6x + 13)

Complete the squares of the denominator to get
(x+3)²+2²
Let y = x+3
dy = dx
and x = y-3

I=∫ 2xdx/(x^2 + 6x + 13)
= ∫ 2(y-3)dy/(y²+2²)
=∫ 2ydy/(y²+2²) - ∫6dy/(y²+2²)
The first one integrates to a log, and the second one to an atan, giving
I=ln(x²+6x+13) - 3atan((x+3)/2) + C