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wolfram-mathematicaintegral

Unexpected behavior of FullSimplify with integers

I executed this command in Mathematica

FullSimplify[Integrate[Sin[x] Sin[k x ], {x, 0, Pi}], Element[k, Integers]]

and got 0. However, if I execute the same command with k=1.

Integrate[Sin[x] Sin[k x] /. {k -> 1}, {x, 0, Pi}]

I get the correct answer, Pi/2. Is this normal? I would expected a conditional based on k=1.

Solution

  • It's a flaw

    The integral:

    Integrate[Sin[x] Sin[k x], {x, 0, Pi}]

    evaluates to:

    -(Sin[k \[Pi]]/(-1 + k^2))

    If you do Simplify[%, Element[k,Integers]], it will assume Sin[k \[Pi]] being zero. The flaw is, Simplify will assume 0 multiplies any non-directly-zero symbol being reduced to 0, even if could be undefined:

    0 / x (*Evaluates to 0*)
ReleaseHold[Hold[0 / x] /. x->0] (*boom*)

    This is explained in detail within Possible Issues section of Simplify.

    All of above applies to FullSimplify