What is the Inverse Fourier Transform

Question

what is the inverse fourier transform for:

$(6w^2 + 20 )/( w^4 + 7w^2 + 12)$

I tried to simplify it and look it up from the table but it didn't work, can someone give me a hint?

Answer 1

Hint:

Use Partial Fraction Decomposition

Writing $w^2=y$

$$\dfrac{6w^2+20}{(w^2+3)(w^2+4)}=\dfrac{6y+20}{(y+3)(y+4)}=\dfrac a{y+3}+\dfrac b{y+4}$$

$\implies6y+20=y(a+b)+3b+4a$

$a=?,b=?$