Integration of rational functions How would you integrate a rational function like this:
$$\frac{2x+3}{x^2+2x+10}$$

$$\int \frac{2x+3}{x^2+2x+10} dx$$
$$= \int \frac{2x+2+1}{x^2+2x+10} dx $$
$= \int \frac{2x+2}{x^2+2x+10} dx + \int \frac{1}{x^2+2x+10}dx$
Let solve them individually,
$$\int \frac{2x+2}{x^2+2x+10} dx$$
$$\int \frac{f'(x)}{f(x)} dx = \log|f(x)|$$
$$= \log |x^2 + 2x + 10| + c$$
Other part -
$$\int \frac{1}{x^2+2x+10} dx$$
Use completing the square -
$$\int \frac{1}{x^2+2x+1+9} dx$$
$$=\int \frac{1}{(x+1)^2+(3)^2} dx$$
$$\int \frac 1{x^2+a^2} dx = \frac 1a tan^{-1}\frac xa + c$$
$$=\frac 13 tan^{-1}\frac {x+1}3+ c$$
Add both to get complete answer.
