This is the question, I have solved it but I need someone to double check my solution. Question: Find the temperature $u(x, t)$ in a rod of length $L$ if the initial temperature is $f (x)$ throughout and if the ends $x=0$ and $x=L$ are insulated. $$F(x) = \begin{cases} x, \ 0<x<1 \\ 0, \ 1<x<2\end{cases}$$ and the length of the rod is $L=2$.
Solution: For an insulated rod the solution $X(x,t)= a_0/2+\sum B_n cos(n\pi x/L)e^{(n^2π^2α^2)t/L}$
I found $a_0= 1$ and $Bn= (−(2/nπ)sin(nπ/2)+(2/nπ)2cos(nπ/2)−(2/nπ)^2)$
then just plug in the coefficients into the sum. I am just not sure if these are the correct values for the $a_0$ and $B_0$ coefficients.
(Note: This looks similar to this question)
(As another user pointed out), your formula for the solution should be written:
$$ X(x,t)= \frac{a_0}{2}+\sum B_{\mathbf{n}} \frac{\cos(n\pi x)}{L}\mathbf{e}^{−\frac{n^2\pi^2\alpha^2}{L}t} $$
Recall $a_0 = (2/L)\int_0^L f(x) dx, \ $ $B_n = (2/L)\int_0^L \cos(n\pi x / L) f(x) dx$. So, with $f(x)$ as above, $L=2$, we have
$$a_0 = (2/2)\int_0^1 x dx = \frac{1}{2}$$
$$b_n = (2/2)\int_0^1 \cos(n\pi x / 2) x dx = [x (2/n\pi) \sin(n \pi x / 2) + (2/n\pi )^2 \cos(n \pi x / 2)]_{x=0}^1$$ $$ = (2/n \pi) \sin(n \pi / 2) + (2/n \pi)^2(\cos(n\pi / 2) - 1) $$ so it looks like you are on the right track, but made a small sign error somewhere.