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Question Number 26559 by abdo imad last updated on 26/Dec/17

$$letputF\left(x\right)={\int }_0^{\mathrm{\infty }}{e}^{-tx}\frac{sint}{t}dtwithx⩾0$$$$weacceptthatFisclass{C}^{1}on[0,\propto [$$$$calculate\frac{\partial F}{\partial x}andfindF\left(x\right)$$$$thenfindthevalueof{\int }_0^{\mathrm{\infty }}\frac{sint}{t}dt$$

Commented by prakash jain last updated on 27/Dec/17

$$\frac{dF}{dx}={\int }_0^{\mathrm{\infty }}\frac{\partial }{\partial x}\left(e^{-tx}\frac{\sin t}{t}\right)dt$$$$=-{\int }_0^{\mathrm{\infty }}{e}^{-tx}\sin tdt$$$$={\left[\frac{e^{-tx}(x\sin t+\cos t)}{x^2+1}\right]}_{t=0}^{t=\mathrm{\infty }}$$$$=-\frac{1}{x^2+1}$$$$F\left(x\right)=-{\tan }^{-1}x+C$$$$F\left(\mathrm{\infty }\right)=0$$$$\Rightarrow C=\frac{\pi }{2}$$$$F\left(x\right)=-{\tan }^{-1}x+\frac{\pi }{2}$$$$F\left(0\right)={\int }_0^{\mathrm{\infty }}\frac{\sin t}{t}dt=\frac{\pi }{2}$$

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