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Question Number 156369 by tabata last updated on 10/Oct/21
solve by betta function ∫_0 ^(2π) sin^5 (z) dz
$${solve}\:{by}\:{betta}\:{function}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} {sin}^{\mathrm{5}} \left({z}\right)\:{dz} \\ $$
Commented by tabata last updated on 10/Oct/21
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Commented by aliyn last updated on 10/Oct/21
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Answered by Mathspace last updated on 11/Oct/21
∫_0 ^(2π)  sin^5 x dx    =∫_0 ^π  sin^5 x dx +∫_π ^(2π ) sin^5 x dx(→x=π+u)  =∫_0 ^(π ) sin^5 xdx+∫_0 ^π (−sin^5 u)du  =0     (no need for betta...!)
$$\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:{sin}^{\mathrm{5}} {x}\:{dx} \\ $$$$ \\ $$$$=\int_{\mathrm{0}} ^{\pi} \:{sin}^{\mathrm{5}} {x}\:{dx}\:+\int_{\pi} ^{\mathrm{2}\pi\:} {sin}^{\mathrm{5}} {x}\:{dx}\left(\rightarrow{x}=\pi+{u}\right) \\ $$$$=\int_{\mathrm{0}} ^{\pi\:} {sin}^{\mathrm{5}} {xdx}+\int_{\mathrm{0}} ^{\pi} \left(−{sin}^{\mathrm{5}} {u}\right){du} \\ $$$$=\mathrm{0}\:\:\:\:\:\left({no}\:{need}\:{for}\:{betta}…!\right) \\ $$$$ \\ $$$$ \\ $$

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