Partial fractions with complex numb

[Miike012]

How do I turn 1/(x⁴+1) into partial fractions?

This is what I did. Let me know if this is correct

1/(x⁴+1) = 1/[(x^2+i)(x^2-i)] = (Ax+B)/(x²+i) + (Cx + D)/(x²-1)

Then I set x = 0

1 = (D-B)i .. My first equation would be D-B = 0.

Is that correct so far?

 

[Infrared]

No need to use complex numbers. $x^4+1=(x^2+1)^2-2x^2=(x^2+1+\sqrt{2}x)(x^2+1-\sqrt{2}x)$

This is a very messy integral to do by hand, by the way.