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Question Number 79580 by john santu last updated on 26/Jan/20

$$\mathrm{does}\mathrm{this}\mathrm{matter}\mathrm{reasonable}$$$$\int \sin {}^{\mathrm{x}}\left(\mathrm{x}\right)\mathrm{dx}?$$

Commented by MJS last updated on 26/Jan/20

$$\mathrm{first}\mathrm{of}\mathrm{all},\mathrm{find}\mathrm{out}\mathrm{where}{\sin }^{x}x\mathrm{is}\mathrm{defined}...$$

Commented by john santu last updated on 26/Jan/20

$$\mathrm{how}\mathrm{to}\mathrm{define}\sin {}^{\mathrm{x}}\left(\mathrm{x}\right)\mathrm{sir}?$$$$-1⩽\sin \left(\mathrm{x}\right)⩽1$$$$0⩽\sin {}^{\mathrm{n}}\left(\mathrm{x}\right)⩽1\mathrm{for}\mathrm{n}\mathrm{even}\mathrm{number}$$$$-1⩽\sin {}^{\mathrm{n}}\left(\mathrm{x}\right)⩽1\mathrm{for}\mathrm{odd}\mathrm{number}$$

Commented by mr W last updated on 26/Jan/20

$$functionisdefinedin2k\pi <x<(2k+1)\pi ?$$

Commented by MJS last updated on 26/Jan/20

$$\mathrm{but}\mathrm{for}\mathrm{the}\mathrm{integral}x\in \mathbb{R}$$$$y={\left(\sin x\right)}^x$$$$\mathrm{let}t=\sin x\Rightarrow -1⩽t⩽1$$$$\iff x=\{\begin{array}{l}2n\pi +\arcsin t\\ (2n+1)\pi -\arcsin t\end{array}$$$$y_n=\{\begin{array}{l}{t}^{2n\pi +\arcsin t}\\ t^{(2n+1)\pi -\arcsin t}\end{array}$$$$y_n\mathrm{is}\mathrm{defined}\mathrm{for}0⩽t⩽1$$$$\mathrm{for}t<0{\mathrm{it}}^{\prime }\mathrm{s}\mathrm{harder}$$

Commented by john santu last updated on 26/Jan/20

$$\mathrm{there}\mathrm{is}\mathrm{no}\mathrm{explanation},\mathrm{at}\mathrm{which}$$$$\mathrm{he}\mathrm{is}\mathrm{defined}$$

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