The problem im given is:
Use Laplace transforms to solve the initial value problem.
$$\ddot x +x=\sin(2t)$$
$$x(0)=0=\dot x(0)$$
I first do the following Laplace transforms:
$$\mathcal{L}\{\ \ddot x(t)\}=s^2X(s),$$
$$\mathcal{L}\{ \sin(2t)\}=\frac{2}{s^2+4}$$
Taking into account the initial values, rewriting the equation as:
$$s^2X(s)+X(s)=\frac{2}{s^2+4}$$
Then solving for $X(s)$:
$$X(s)=\frac{2}{(s^2+1)(s^2+4)}$$
Then here is where i get stuck, in that i dont understand how to correctly use the method of partial fractions on this problem, to get to a point where i could do an inverse Laplace transform or if that's the right approach?
$$\frac{2}{(s^2+1)(s^2+4)}=\frac{A}{s^2+1}+\frac{B}{s^2+4}$$
You're definitely on the right track here. Let's solve your system. If we multiply both sides by $(s^2+1)(s^2+4)$, we see that
$$2 = A(s^2+4) + B(s^2+1).$$
Or equivalently
$$2 = (A+B)s^2 + (4A+B).$$
By equating coefficients (which can be made rigorous by differentiating both sides sufficiently many times), we see that $A+B = 0$ since there is no $s^2$ term on the left hand side and $4A+B = 2$. Can you take it form here?
As a note for future posts, if you put two sets of two dollar signs around your LaTeX code, it will render it as an honest-to-goodness equation (not just inline math) which helps readability a lot.