integrate 1/(x^2+4x+13)

The first step is to notice that this is a standard integral in the form of 1/(x^2+a^2). In order to reach this form, we must first complete the square. Then we have 1/(x+2)^2-4+13=1/(x+2)^2+9. We can then use the substitution u = x+2. du=dx to obtain 1/u^2+3^2, our required form. Using a formula booklet, we see that this integrates into 1/3 arctan(u/3). We then substitute for u giving 1/3 arctan(x+2)/3