How do you use partial fraction decomposition to decompose the fraction to integrate #x/(x^2+4x+13)#?
This function should not be integrated with partial fractions, but with a substitution (after completing the square in the denominator). The final answer is
The denominator cannot be factored without resorting to complex numbers, so we complete the square in the denominator instead.
Completing the square for the denominator
The integral can therefore be written as
The first of these integrals can be done by inspection to see that
The second integral can be done by a bit of tricky algebra, motivated by the fact that
Hence, after combining the integration constants into just one, the final answer is