## Calculus 10th Edition

The integral is: $$\frac{1}{3}\arctan\frac{x+2}{3}+c$$ where $c$ is arbitratry. The graph of the antiderivatives is on the following figure.
Using Wolfram Mathematica (which is an example of computer algebra system) with the code y = Integrate[1/(x^2 + 4 x + 13), x] Plot[{y + 2, y - 1}, {x, -10, 10}] We get that integral $$\int\frac{1}{x^2+4x+13} dx=\frac{1}{3}\arctan\frac{x+2}{3}+c.$$ We get two antiderivatives for two different choices for $c$. We have chosen $c=2$ (blue) and $c=-1$ (orange) and they are plotted on the graph.