Integrating sin(m x) against cos(n x).

Manipulate[{Plot[{Sin[m x],Cos[n x]},{x,-Pi,Pi},PlotLabel->"Sine and Cosine"],Plot[Sin[m x]*Cos[n x],{x,-Pi,Pi},PlotLabel->"Sin times Cosine"]},{{m,1},1,10,1,Appearance->"Open"},{{n,1},1,10,1,Appearance->"Open"}]

SinCosTable=Table[Integrate[Sin[m x]*Cos[n x],{x,-Pi,Pi}],{m,1,10},{n,1,10}]
Print["(∫_(-π)^π)Sin[mx]Cos[nx]\[DifferentialD]x, for m down rows, n across columns"]
TableForm[SinCosTable,TableHeadings->Automatic]

Integrating sin(m x) against sin(n x)

Manipulate[{Plot[{Sin[m x],Sin[n x]},{x,-Pi,Pi},PlotLabel->"Sin[mx] and Sine[nx]"],Plot[Sin[m x]*Sin[n x],{x,-Pi,Pi},PlotLabel->"Sin times Sine"],Integrate[Sin[m x]Sin[n x],{x,-Pi,Pi}]},{{m,1},1,10,1,Appearance->"Open"},{{n,1},1,10,1,Appearance->"Open"}]

SinSinTable=Table[Integrate[Sin[m x]*Sin[n x],{x,-Pi,Pi}],{m,1,10},{n,1,10}]
Print["(∫_(-π)^π)Sin[mx]Sin[nx]\[DifferentialD]x, for m down rows, n across columns"]
TableForm[SinSinTable,TableHeadings->Automatic]

Integrating Cos[m x] against Cos[n x]

Manipulate[{Plot[{Cos[m x],Cos[n x]},{x,-Pi,Pi},PlotLabel->"Cos[mx] and Cos[nx]"],Plot[Cos[m x]*Cos[n x],{x,-Pi,Pi},PlotLabel->"Sin times Sine"],Integrate[Cos[m x]*Cos[n x],{x,-Pi,Pi}]},{{m,1},1,10,1,Appearance->"Open"},{{n,1},1,10,1,Appearance->"Open"}]

CosCosTable=Table[Integrate[Cos[m x]*Cos[n x],{x,-Pi,Pi}],{m,1,10},{n,1,10}]
Print["(∫_(-π)^π)Cos[mx]Cos[nx]\[DifferentialD]x, for m down rows, n across columns"]
TableForm[SinSinTable,TableHeadings->Automatic]

?Control

f[x_]:=x^2
a=(1/(2Pi))Integrate[f[x],{x,-Pi,Pi}]
b1=(1/Pi)Integrate[f[x]*Sin[x],{x,-Pi,Pi}]
c1=(1/Pi)Integrate[f[x]*Cos[x],{x,-Pi,Pi}]
b2=(1/Pi)Integrate[f[x]*Sin[2x],{x,-Pi,Pi}]
c2=(1/Pi)Integrate[f[x]*Cos[2x],{x,-Pi,Pi}]

g[x_]:=a+b1 Sin[x]+c1 Cos[x]
g[x]
h[x_]:=g[x]+b2*Sin[2x]+c2*Cos[2x]
h[x]
Plot[{f[x],g[x],h[x]},{x,-Pi,Pi}]

Limit[(1+1/n^2)^n,n->Infinity]
Limit[(1+1/n)^(n^2),n->Infinity]

N[Sqrt[1/2]]
N[Sqrt[1.5]]