Assumptions not working with Trianglewave or Floor functions

Question

The assumption below:

Simplify[Integrate[Sin[2 n π x], {x, 0, 1}], Assumptions -> n ∈ Integers]

simplifies the integration and gives $0$ as the correct answer. However, for other functions like TriangleWave[x] or Floor[x], assumptions do not appear to work (Linux, Version 10.0).

None of the commands below simplifies:

Simplify[Integrate[TriangleWave[n x]^2, {x, 0, 1},   Assumptions -> n ∈ Integers]]
Simplify[Integrate[TriangleWave[n x]^2, {x, 0, 1}],  Assumptions -> n ∈ Integers]
Simplify[Integrate[Floor[x], {x, 0, n}],  Assumptions -> {n ∈ Integers, n > 0}]

I have to use command like:

Table[Integrate[TriangleWave[n x]^2, {x, 0, 1}], {n, 1, 5, 1}]

to generate a list of result, and "extract" the formula $f(n)$ by observation. Did I miss something?

Answer 1

Although it does not help in these particular cases, it is generally better to use Assuming as in Assuming[{assumptions}, Simplify[Integrate[ ... ]]] since the assumptions then will be used in both the Integrate and Simplify steps.

A workaround since Mathematica only solves the integrals for specific values of n is to generate a sequence for sequential values of n and then use FindSequenceFunction to find the closed-form expression for general n.

FindSequenceFunction[
 Table[
  Integrate[
   TriangleWave[n x]^2,
   {x, 0, 1}],
  {n, 10}],
 n]

(*  1/3  *)

FindSequenceFunction[
 Table[
  Integrate[
   Floor[x],
   {x, 0, n}],
  {n, 10}],
 n]

(*  1/2 (-1 + n) n  *)