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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