$\displaystyle \int \dfrac{x + 3}{x^{2} + 2x + 2}dx$

$\displaystyle \dfrac{1}{2} \int \dfrac{(x + 1) 2dx}{(x + 1)^{2} + 1} + \int \dfrac{2dx}{(x + 1)^{2} + 1}$ - What's going on with the numerator? I understand the denominator (completing the square).

For instance, how can $\displaystyle dx$ be fixed (u substitution)? Given: $\displaystyle u = x^{2} + 2x + 2$ and $\displaystyle du = 2x + 2$. Note: $\displaystyle du$ and $\displaystyle dx$ have to be fixed before splitting the fraction into two fractions.

$\displaystyle \dfrac{1}{2} \ln(x^{2} + 2x + 2) + 2 \arctan(x + 1) + C$ - Final answer