How do you use partial fraction decomposition to decompose the fraction to integrate #x/(x^2+4x+13)#?

1 Answer
Jun 24, 2015

This function should not be integrated with partial fractions, but with a substitution (after completing the square in the denominator). The final answer is #int x/(x^2+4x+13)\ dx#

#=1/2 ln(x^2+4x+13)-2/3 arctan((x+2)/3)+C#

Explanation:

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 #x^2+4x+13# can be done by taking the coefficient 4 of #x#, dividing it by 2 and squaring that to get 4, and then writing the denominator as #x^2+4x+4+13-4=(x^2+4x+4)+9=(x+2)^2+9#.

The integral can therefore be written as #\int x/(x^2+4x+13)\ dx=\int x/((x+2)^2+9)\ dx#.

Now let #u=x+2# so that #x=u-2# and #dx=du# to write

#\int x/((x+2)^2+9)\ dx=\int u/(u^2+9)\ du-\int 2/(u^2+9)\ du#

The first of these integrals can be done by inspection to see that #\int u/(u^2+9)\ du=1/2 ln|u^2+9|+C=1/2 ln(x^2+4x+13)+C# (you could also do another substitution #w=u^2+9#, #dw=2u\ du#, and #u\ du = 1/2 dw# to help you do this). Note also that #x^2+4x+13=(x+2)^2+9>0# for all #x\in RR#.

The second integral can be done by a bit of tricky algebra, motivated by the fact that #\int 1/(1+x^2)\ dx=arctan(x)+C#. Here's the algebra:

#\int 2/(u^2+9)\ du=2/9 \int 1/(1+(u/3)^2)\ du=2/3 arctan(u/3)+C#

#=2/3 arctan((x+2)/3)+C# (you could also do another substitution #w=u/3#, #du=3dw# to help you do this). Note at the end that we have replaced #u# by #x+2#.

Hence, after combining the integration constants into just one, the final answer is

#int x/(x^2+4x+13)\ dx#

#=1/2 ln(x^2+4x+13)-2/3 arctan((x+2)/3)+C#