int (x^2-1)/(x^4+x^2+1) dx?
2 Answers
Explanation:
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Explanation:
Note that:
x^4+x^2+1 = (x^2-x+1)(x^2+x+1)
We find:
(x^2-1)/(x^4+x^2+1) = (x-1/2)/(x^2-x+1)-(x+1/2)/(x^2+x+1)
color(white)((x^2-1)/(x^4+x^2+1)) = 1/2((2x-1)/(x^2-x+1))-1/2((2x+1)/(x^2+x+1))
color(white)((x^2-1)/(x^4+x^2+1)) = 1/2((d/(dx)(x^2-x+1))/(x^2-x+1))-1/2((d/(dx)(x^2+x+1))/(x^2+x+1))
So:
int (x^2-1)/(x^4+x^2+1) dx = int 1/2((2x-1)/(x^2-x+1))-1/2((2x+1)/(x^2+x+1)) dx
color(white)(int (x^2-1)/(x^4+x^2+1) dx) = 1/2 ln abs(x^2-x+1)-1/2ln abs(x^2+x+1) + C