partial fraction expansion...

Question

I tried partial fraction expansion in this way,

but it's just too cumbersome to solve in a test.

Is there another way?

Answer 1

Hint:

$$2s=(a_1+a_2s)(s^2-s+4)+(a_3+a_4s)(s^2+1)$$

Comparing the constant terms $0=4a_1+a_3\iff a_3=?$

Comparing the coefficients of $s^2,0=a_1-a_2+a_3\iff a_2=a_1+a_3=a_1-4a_1=?$

Comparing the coefficients of $s^3, 0=a_2+a_4\iff a_4=-a_2=?$

Comparing the coefficients of $s, 2=-a_1+4a_2+a_4$

Replace the values of $a_2,a_4$ with $a_1$