How do you integrate $(x+2)/(x^2+8x+16)$ using partial fractions?

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Answer 1 · 2016-12-25T23:02:47

$int (x+2)/(x^2+8x+16) dx= ln|x+4|+2/(x+4)+C$

First you factorize the denominator of the rational function:

$x^2+8x+16 = (x+4)^2$

Now you can write:

$(x+2)/(x^2+8x+16) = (x+2)/((x+4)^2)= (x+4-2)/((x+4)^2)=1/(x+4)-2/((x+4)^2)$

The integral is then:

$int (x+2)/(x^2+8x+16) dx= int (dx)/(x+4)-2int(dx)/((x+4)^2)=ln|x+4|+2/(x+4)+C$

How do you integrate (x+2)/(x^2+8x+16)(x+2)/(x^2+8x+16) using partial fractions?