1)

> a0:=int(x^3,x=0..1);
                                   a0 := 1/4

> an:=2*int(x^3*cos(n*Pi*x),x=0..1);
         3   3                2   2
        n  Pi  sin(n Pi) + 3 n  Pi  cos(n Pi) - 6 cos(n Pi) - 6 n Pi sin(n Pi)
an := 2 ----------------------------------------------------------------------
                                         4   4
                                        n  Pi

         12
     + ------
        4   4
       n  Pi

> an:=simplify(subs(sin(n*Pi)=0,an));

                            2   2
                           n  Pi  cos(n Pi) - 2 cos(n Pi) + 2
                   an := 6 ----------------------------------
                                          4   4
                                         n  Pi


              /infinity                                                     \
              | -----   /    2   2                                         \|
              |  \      |  (n  Pi  cos(n Pi) - 2 cos(n Pi) + 2) cos(n Pi x)||
   F := 1/4 + |   )     |6 ------------------------------------------------||
              |  /      |                        4   4                     ||
              | -----   \                       n  Pi                      /|
              \ n = 1                                                       /


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