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

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Answer 1 · 2015-09-26T03:46:33

See the explanation section, below.

$x^2/(x^2+x+4) = ((x^2+x+4)-(x+4))/(x^2+x+4)$

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

Now, the derivative of the denominator is $2x+1$ , so the integral of the fraction is not an $ln$ , but we can make it one:

$x^2/(x^2+x+4)= 1-(x+1/2)/(x^2+x+4)-(7/2)/(x^2+x+4)$

The integral of the first is $x$ , the second is $kln(x^2+x+4)$ and the third is some $tan^-1$ (Complete the square to find the integral.)

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