Can I plug imaginary unit to solve partial fraction?

Question

for example, I have partial fraction to solve:

$$ 1/(x^2+1)(x^2+4)=A/(x^2+1)+B/(x^2+4)\ $$

then I need to solve $$ 1=A(x^2+4)+B(x^2+1) $$

Can I plug x=i and 2i so that I can get the value of A and B directly?

Is there any logical error to do that?

Answer 1

If it's meant for integration, it's easier to put $x^2=y$

$$\dfrac1{(y+1)(y^2+1)}=\dfrac A{1+y}+\dfrac{By+C}{1+y^2}$$

$$\iff1=A(1+y^2)+(By+C)(1+y)$$

$A+B=B+C=0,C+A=1\implies-B=C=A=\dfrac12$

Can you take it from here?