For the heat equation on a semi infinite domain we have been shown the method of images. Which to my understanding is basically using an even or odd extension to match the boundary condition.ie for

we use the even extension which meets this condition.And when the derivative is not specified but instead we use the odd extension which has non zero derivative at x=0 but meets this boundary condition.

For example:

Then using the general solution to solve this i get

The solution does not have a two. Which implies that the initial condition has been extended to an infinite domain and there is only one integral. As we were shown in class i get two integrals and since -sin(-x)=sinx i get the factor of two. I am wondering why there isnt a two in the solution.