ClearAll["Global`*"]

R2=((w L ((L/2)-L1))/(L2-L1));R1=w L - R2;
Ma[x_]=((-w (x^2))/2);Mb[x_]=((-w (x^2))/2)+R1*(x-L1);Mc[x_]=((-w (x^2))/2)+R1*(L2-L1)+w L(x-L2);  (* we did all this stuff in class alrady *)

yprimeA[x_]=((Integrate[Ma[x],x])/(Elast I0))+c1a;
yprimeB[x_]=((Integrate[Mb[x],x])/(Elast I0))+c1b;
yprimeC[x_]=((Integrate[Mc[x],x])/(Elast I0))+c1c;
(* You can see that I added the constants of integration by hand! *)

yA[x_]=Integrate[yprimeA[x],x]+c2a;
yB[x_]=Integrate[yprimeB[x],x]+c2b;
yC[x_]=Integrate[yprimeC[x],x]+c2c;

(* We have 6 new constants, so we need 6 BC's! ... *)

(* yA[L1] = 0, yB[L1]=0, yB[L2]=0, and yC[L2]=0      value fixed at supports   *)
(* y'A[L1]=y'B[L1], y'B[L2]=y'C[L2]                  slopes match at supports  *)

constants=Solve[{yA[L1]==0 && yB[L1]==0 && yB[L2]==0 && yC[L2]==0 && yprimeA[L1]==yprimeB[L1] &&yprimeB[L2]==yprimeC[L2]},{c1a,c2a, c1b,c2b,c1c,c2c}]//Flatten;

c1a=c1a/.constants;
c1b=c1b/.constants;
c1c=c1c/.constants;
c2a=c2a/.constants;
c2b=c2b/.constants;
c2c=c2c/.constants;
yA[x]//Simplify
yB[x]//Simplify
yC[x]//Simplify

(* Let's put in some numbers so we can make a plot... *)
(* I'm using pounds & inches for all units *)

I0 = 21;Elast = 10*(10^6); w = 500;
L=40;L1=10;L2=25;  (* i.e., the supports are a little on the left side *)
dymax =Min[yA[0],yC[L],yB[L/2]]; (* attempt to determine the vertical plot scaling *)
yA[x]
yB[x]
yC[x]
plotA=Plot[yA[x],{x,0,L1},PlotRange->{{0,L},{-dymax,dymax}}];
plotB=Plot[yB[x],{x,L1,L2},PlotRange->{{0,L},{-dymax,dymax}}];
plotC=Plot[yC[x],{x,L2,L},PlotRange->{{0,L},{-dymax,dymax}}];
Show[plotA,plotB,plotC,Frame->True]