# Partial fraction with same denominator

Is the following fraction (actually a Laplace transform) a kind of partial fraction?

$$\frac{4s+3}{{s^2}+3}$$

Can this be solved this way? $$\frac{A}{s}+\frac{B}{s+{\frac{3}{s}}}$$

If not can you please tell me how to find inverse transform?

If you want to keep everything real, this is already decomposed into partial fractions. For the inverse Laplace transform, just split it as $$\frac{4s+3}{s^2+3} = 4 \frac{s}{s^2+3} + 3\frac{1}{s^2+3}.$$ You should be able to invert each term separately.
• Ok, if the first one had 9 instead of 3 I would say it is $\cos(3t)$ but what is it in this case? Jun 28 '12 at 19:07
• The inverse Laplace transform of $s/(s^2 + a^2)$ is $\cos(at)$. If $3 = a^2$, what do you suppose $a$ is? Jun 28 '12 at 19:09
If you don't need to keep it real, the roots of $s^2 + 3$ are $\pm \sqrt{3} i$, and the partial fraction decomposition is $$\frac{4s+3}{s^2+3} = {\frac {2-i\sqrt {3}/2}{s-i\sqrt {3}}}+{\frac {2+i\sqrt {3}/2}{ s+i\sqrt {3}}}$$