Question

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

Answer 1

See the explanation section, below.

Explanation:

`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.)