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Question Number 142028 by mnjuly1970 last updated on 25/May/21

$$............Calculus.........$$$${\int }_0^{\mathrm{\infty }}\frac{sin\left(sin\left(x\right)\right).e^{cos\left(x\right)}}{x}dx=???$$$$............m.n.....$$

Answered by mindispower last updated on 25/May/21

$$=Im{\int }_0^{\mathrm{\infty }}\frac{e^{cos\left(x\right)+isin\left(x\right)}}{x}$$$$Im{\int }_0^{\mathrm{\infty }}\frac{e^{e^{ix}}}{x}dx=Im{\int }_0^{\mathrm{\infty }}\sum _{k⩾0}\frac{e^{ikx}}{k!x}dx$$$$=\sum _{k⩾0}{\int }_0^{\mathrm{\infty }}\frac{sin\left(kx\right)}{x.k!}dx=\sum _{k⩾0}\frac{1}{k!}{\int }_0^{\mathrm{\infty }}\frac{sin\left(kx\right)}{kx}d\left(kx\right)$$$$=\sum _{k⩾0}\frac{1}{k!}.{\int }_0^{\mathrm{\infty }}\frac{sin\left(x\right)}{x}dx=\frac{\pi }{2}\sum _{k⩾0}\frac{1}{k!}=\frac{\pi }{2}e$$

Commented by mnjuly1970 last updated on 25/May/21

$$zendehbashidtashakor.thanksalot$$$$mrpower...$$

Commented by mindispower last updated on 25/May/21

$$withepleasursirthankyou$$

Answered by Dwaipayan Shikari last updated on 25/May/21

$$e^{cosx}sin\left(sinx\right)=\underset{n=0}{\sum ^{\mathrm{\infty }}}\frac{sin\left(nx\right)}{n!}$$$${\int }_0^{\mathrm{\infty }}\underset{n=0}{\sum ^{\mathrm{\infty }}}\frac{sin\left(nx\right)}{xn!}dx=\underset{n=0}{\sum ^{\mathrm{\infty }}}\frac{1}{n!}{\int }_0^{\mathrm{\infty }}\frac{sin\left(nx\right)}{x}dx=\frac{\pi e}{2}$$

Commented by mnjuly1970 last updated on 26/May/21

$$thanksalot...$$

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