Scratch work: integrals of products of sine functions

Integrate[Sin[3 π x]Sin[6 π x],{x,-1,1}]

Plotting the Fourier series for the odd 2-periodic extension of f(x)=1-x

Manipulate[
Plot[
{1-x,Sum[(2/(k π))Sin[k π x],{k,1,max}]},
{x,-5,5}],
{max,1,50,1}]

Plotting the Fourier series for the even 2-periodic extension of f(x)=1-x

Manipulate[
Plot[
{1-x,(1/2)+Sum[((2(1-(((-1))^k)))/((k^2) (π^2)))Cos[k π x],{k,1,max}]},
{x,-5,5}],
{max,1,50,1}]

Complex Fourier Series Demo for f(x)=(e^x)

General formula for the complex Fourier coefficients:

c[k_]:=(((-1))^k)(((Exp[π]-Exp[-π]))/(2π(1-I*k)))

Compute some complex Fourier coefficients for indexes k from -10 to 10:

c/@Range[-10,10]

Plot the complex Fourier series for (e^x). Note that this Fourier series is 2π-periodic.

Manipulate[
Plot[
{Exp[x],Sum[c[k]*Exp[I*k*x],{k,-max,max}]},
{x,-5,5}],
{max,1,50,1}]