How do you integrate #int (x+4)/(x^2 + 2x + 5)dx# using partial fractions?

1 Answer
benjidunn
Apr 1, 2016

#1/2[ln|x^2+2x+5|+3arctan((x+1)/2)]+c#

Explanation:

First split the integral into two parts:

#x+1# is a scalable factor of the derivative of #x^2+2x+5#, so divide #x+4# by #x+1#

#int(x+4)/(x^2+2x+5)dx=int(x+1)/(x^2+2x+5)dx+int3/(x^2+2x+5)dx#

Algebraic manipulation yields:
#1/2int(2x+2)/(x^2+2x+5)dx+3int1/((x+1)^2+4)dx#
#=1/2ln|x^2+2x+5|+3/2arctan((x+1)/2)+c#
#=1/2[ln|x^2+2x+5|+3arctan((x+1)/2)]+c#

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