Question

integrate x+2/x^2+2x+3

Answer 1

complete the square:
\[\int\limits \frac{x+2}{(x+1)^2 +2} dx\]
u = x+1
du = dx
\[\int\limits \frac{u+1}{u^2 +2} du\]
split into 2 integrals
\[\int\limits \frac{u}{u^2 +2} + \int\limits \frac{1}{u^2 +2}\]
first integral:
w = u^2 +2
dw = 2u du
\[\frac{1}{2} \int\limits \frac{dw}{w} = \frac{1}{2} \ln w\]
2nd integral:
u = sqrt2 tan w
du = sqrt2 sec^2 w
\[\int\limits \frac{1}{2 \sec^2 w} \sqrt{2} \sec^2 w = \frac{\sqrt{2}}{2} w\]
plugging back in for x
\[\rightarrow \ln (x^2 +2x+3) + \frac{\sqrt{2}}{2} \tan^{-1} (\frac{x+1}{\sqrt{2}}) + C\]