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Which of the following is true for all possible non-zero choices of integers $m,n;m\neq n,$ or all possible non-zero choices of real numbers $p,q;p\neq q,$ as applicable?

  1. $\displaystyle{}  \dfrac{1}{\pi }\int_{0}^{\pi }\sin m\theta \:\sin n\theta d\theta =0 \\$
  2. $ \displaystyle{}  \dfrac{1}{2\pi }\int_{\frac{-\pi }{2}}^{\frac{\pi }{2} }\sin p\theta \:\sin q\theta d\theta =0 \\$
  3. $\displaystyle{}  \dfrac{1}{2\pi }\int_{-\pi }^{\pi }\sin p\theta \:\cos q\theta d\theta =0 \\$
  4. $\displaystyle{} \lim_{\alpha \rightarrow \infty }\dfrac{1}{2\alpha }\int_{-\alpha }^{\alpha }\sin p\theta \:\sin q\theta d\theta =0$
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