(∫_(-(L/2))^(L/2))((2 (sin^2)(((π (2 m+1) x)/L)))/L)\[DifferentialD]x

(∫_(-(L/2))^(L/2))((2 (cos^2)(((π (2 m) x)/L)))/L)\[DifferentialD]x

FullSimplify[Sin[ Pi (2 m )/2], m ∈ Integers ]

FullSimplify[Cos[ Pi (2 m +1)/2], m ∈ Integers ]