Integrate[(2/30)*(40-3*x)*Sin[((n*Pi*x)/30)],{x,0,30}]

c[n_] = ((20 (4 n π+5 n π *(-1)^n))/((n^2) (π^2)))

w[x_,t_] = Sum[c[n]*Exp[-((n^2*Pi^2)/(30^2))*t]*Sin[((n*Pi*x)/30)],{n,1,200}];

u[x_,t_] = 20+x + w[x,t];

Manipulate[
Plot[u[x,t],{x,0,30},PlotRange->{{0,30},{0,60}}],
{t,0,100}]