# Orthogonality Sin(nx)Sin(mx)

I want to prove that $$\int_{0}^{\pi } \sin(n \, x) \, \sin(m \, x) \, \mathrm{d} x=0$$ for m,n integers and $$m\neq n$$ My try is:

$$Assumptions = m ≠ n$$Assumptions = m ∈ Integers
$Assumptions = n ∈ Integers Integrate[Sin[m x] Sin[n x], {x, 0, π} ]  But no luck ## 1 Answer You overwrite $Assumptions several times, so it does not contain all the information that you meant to provide. Also Simplify can help where Integrate gave up the simplification. Try this:

$Assumptions = m != n && m \[Element] Integers && n \[Element] Integers Simplify@Integrate[Sin[m x]*Sin[n x], {x, 0, \[Pi]}]  • Great, thanks!! – Alicia Roberts Dec 5 '19 at 11:07 • You're welcome! – Henrik Schumacher Dec 5 '19 at 11:07 • Also $Assumptions = m \[Element] Integers && n \[Element] Integers gives the same result 0 which is obviously wrong for the case n==m . Still (v12) a Mathematica problem? – Ulrich Neumann Dec 5 '19 at 11:17
• Uh, yes. Must be the thing about Simplify assuming some generacity of the symbols... =/ – Henrik Schumacher Dec 5 '19 at 11:18
• @UlrichNeumann Funny, I was making the same comment at the same time under the OP. :) – Michael E2 Dec 5 '19 at 11:23