# problem solving this integral of half line dirichlet heat equation

I am studying heat equations and I am trying to categorize the questions and their general solution as well as techniques to solve them. I reached to this excercise:
for half line $$0
$$u_t-u_{xx}=0$$
$$u(x,0)=sin(2x)$$
$$u(0,t)=0$$
I solved the problem to some point but I am stuck with this integral. I do not know how to solve this:
$$\int_{-\infty}^{\infty} e^{\frac{-(x-y)^2}{4t}}sin(2y) \,dy$$ I would appreciate any help with this

Separation of variables might make you guess: $$u(x,t)= e^{-4t}\sin(2x)$$