I can see that $\displaystyle L^{-1}\{\frac{1}{s(s^2+5)}\}=\frac{1}{5}-\frac{1}{5}cos(\sqrt{5}t)$,

because,

$\displaystyle L\{\int_0^tf(\tau)d\tau\}=\frac{F(s)}{s}$

But when I try to use partial fractions $\displaystyle L^{-1}\{\frac{1}{s(s^2+5)}\}=L^{-1}\{\frac{A}{s}+\frac{B}{s^2+5}\}$

$\displaystyle A(s^2+5)+Bs=1$

$\displaystyle s=0\rightarrow A=\frac{1}{5}$

$\displaystyle As^2+Bs+5A=1$

This doesn't seem to work at all. Basically this gives inconsistent values of A and a value of zero for B. Am I approaching the method of partial fractions wrong? Since there isn't a squared term on the right, A must be zero right?

EDIT: LOL, I know where I'm going wrong. I need to use As+B in the denominator because the one factor is quadratic. Sorry for wasting time.