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Sin[(m - n)*x]/(2*(m - n)) - Sin[(m + n)*x]/(2*(m + n))

Integrate[Sin[m*x]*Sin[n*x], x] ==
Sin[(m - n)*x]/(2*(m - n)) - Sin[(m + n)*x]/ (2*(m + n)) 

Integrate[Sin[m x] Sin[n x], x] == 

Sin[(m - n) x]   Sin[(m + n) x]
-------------- - --------------
  2 (m - n)        2 (m + n)
Time to compute: 0.13 second
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