Σ_(k=0) ^(p−1) ((p),(k) )sin [2(p−k)x]=? ((p),(0) )sin (2px)+ ((p),(1) )sin [(2p−2)x]+ ((p),(2) )sin [(2p−4)x]+...+ ((( p)),((p−1)) )sin (2x)=2^p ∙cos^p (x)∙sin (px) ??? or ∫_0 ^∞ ((cos^p (x)∙sin (px))/x)dx=(π/2)(1−2^(−p) ) why ???


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Question Number 140282 by qaz last updated on 06/May/21

$\underset{k = 0}{\sum^{p - 1}} (\begin{matrix} p \\ k \end{matrix}) \sin [2 (p - k) x] = ?$ $(\begin{matrix} p \\ 0 \end{matrix}) \sin (2 p x) + (\begin{matrix} p \\ 1 \end{matrix}) \sin [(2 p - 2) x] + (\begin{matrix} p \\ 2 \end{matrix}) \sin [(2 p - 4) x] + . . . + (\begin{matrix} p \\ p - 1 \end{matrix}) \sin (2 x) = 2^{p} \cdot \cos^{p} (x) \cdot \sin (p x) ? ? ?$ $o r \int_{0}^{\infty} \frac{\cos^{p} (x) \cdot \sin (p x)}{x} d x = \frac{π}{2} (1 - 2^{- p}) w h y ? ? ?$
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