SetDirectory[$HomeDirectory<>"/Documents/handouts/math467/DiffInt"]

Integrate[(x^3)Sin[n x],{x,0,π},Assumptions->{n∈Integers}](2/π)

Integrate[(x^3)Sin[n x],{x,-π,π},Assumptions->{n∈Integers}](1/π)

%/.{Sin[n π]->0,Cos[n π]->(((-1))^n)}

Expand[%*%]/.(((-1))^(2n))->1

Sum[1/(n^2),{n,1,Infinity}]

Sum[1/(n^4),{n,1,Infinity}]

Integrate[(x^6),{x,-π,π}]/(2π)

%+24(π^2)(π^4)/90-2(π^4)(π^2)/6

%/72

Simplify[%]

Sum[1/(n^6),{n,1,Infinity}]