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Question Number 209923 by OmoloyeMichael last updated on 26/Jul/24

$$\mathit{S}\mathit{o}\mathit{l}\mathit{v}\mathit{e}:\int \frac{\mathit{s}\mathit{i}\mathit{n}(\mathit{x}!)}{\mathit{x}!}\mathit{d}\mathit{x}$$

Answered by MrGaster last updated on 03/Feb/25

$$\int \frac{\sin (x!)}{x!}\mathrm{d}x=\underset{k=0}{\sum ^{\mathrm{\infty }}}{\int }_{k\pi }^{(k+1)}\frac{\sin \left(x\right)!}{x!}$$$$=\underset{k=0}{\sum ^{\mathrm{\infty }}}\left[{(-1)}^k{\int }_0^{\pi }\frac{\sin ((k\pi +t)!)}{(k\pi +t)!}dt\right)$$$$=\underset{k=0}{\sum ^{\mathrm{\infty }}}\left[{(-1)}^k\left(\frac{\cos ((k\pi -t)!)}{(k\pi +t)!}\right){\mid }_0^{\pi }\right]$$$$=\underset{k=0}{\sum ^{\mathrm{\infty }}}\left[{(-1)}^k(\frac{\cos ((k+1)\pi !)}{(k+1)\pi !}-\frac{\cos (k\pi !)}{k\pi })\right]$$$$=\underset{n\to \mathrm{\infty }}{\lim }[{(-1)}^n\frac{\cos ((n-1)\pi !)}{(n+1)!}+\underset{k=1}{\sum ^n}\frac{{(-1)}^{k=1}}{(k-1)\pi !}+\frac{1}{\pi !}]$$$$=\frac{1}{\pi !}+\underset{k=1}{\sum ^{\mathrm{\infty }}}\frac{{(-1)}^{k-1}}{(k-1)\pi !}$$$$I=\int \frac{\sin (x!)}{x!}\mathrm{d}x=\frac{1}{\pi !}+\underset{k=1}{\sum ^{\mathrm{\infty }}}\frac{{(-1)}^{k-1}}{(k-1)\pi !}$$

Commented by mr W last updated on 23/Mar/25

$$butthequestion{didn}^{\prime }task$$$${\int }_0^{\mathrm{\infty }}\frac{\sin (x!)}{x!}\mathrm{d}x$$

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