Solving Heat equation

[Yuchen Wu]

Dear Prof Ivrii,
I did some home assignment for heat equation problems to prepare the quiz, and I found some integrals seems hard to compute. So I am wondering if we need to compute the integral completely correct to get the full mark when do the quiz, or just simplify the heat formula is fine? Thank you.

[Victor Ivrii]

I will post solutions to show that all the integrals can be computed using elementary functions and $\operatorname{erf}(.)$