I am solving a question and I have determined the final solution to be the following $$ u(x,t) = A_0 + \sum_{n=1}^{\infty} A_n \cos(2n\pi x) \exp(-36n^2\pi^2 t) $$
But now I am not quite sure how to solve this equation, it seems like I need to compute the Fourier series expansion
I have computed the following but now am looking for help with the execution, thanks!
$$ A_0 = \int_0^1 \sin (\pi x) dx = \frac{2}{\pi } \approx 0.63622 $$ $$ A_n = 2\int_0^1 \sin(\pi x) \cos(2n\pi x) dx = \frac{4cos^2(\pi n)}{\pi - 4\pi n^2}$$
I've answered your previous question here. There is no need to do anything else, you're done.
The final solution is in the form of a series
$$ u(x,t) = \frac{2}{\pi} + \frac{4}{\pi}\sum_{n=1}^\infty \frac{\cos^2(n\pi)}{1-4n^2}\cos(2n\pi x)\exp(-36n^2\pi^2 t) $$
Note that your $A_n$ were derived using the boundary condition $u(x,0) = \sin(\pi x)$