Can you show steep by step how to do 4pi2 xy sinmx sinny dz

Can you show steep by step how to do

Solution

-( cos(mx+ny) = cos(mx).cos(ny) -sin(mx).sin(ny) )
cos(mx-ny) = cos(mx).cos(ny) + sin(mx).sin(ny)
+
---------------------------------------------------------------------
cos(mx-ny) - cos(mx+ny) = 2 sin(mx).sin(ny)


Integrate[
Integrate[(4/(Pi)^2)*((x + y) Sin[m*x] Sin[n*y]), {x, 0, Pi}], {y, 0,
Pi}]
integration by parts

(4/ (mn)^2) (n Sin[m ] - n Cos[n ] (m \\[Pi] + Sin[m ]) +
m Cos[m ] (-n ] + 2 n ] Cos[n ] - Sin[n ]) +
m Sin[n ])