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Question Number 174770 by Mastermind last updated on 10/Aug/22

$$\int \left(\frac{\mathrm{sinx}}{\sqrt{\mathrm{x}}}\right)\mathrm{dx}$$$$$$$$\mathrm{Mastermind}$$

Answered by Frix last updated on 10/Aug/22

$$\int \frac{\sin x}{\sqrt{x}}dx\stackrel{(\ast )}{=}\sqrt{2\pi }\int \sin \frac{\pi u^2}{2}du\mathrm{which}\mathrm{is}\mathrm{the}$$$$\mathrm{Fresnel}-\mathrm{Integral}=\sqrt{2\pi }\mathrm{S}\left(u\right).\mathrm{the}\mathrm{answer}\mathrm{is}$$$$\int \frac{\sin x}{\sqrt{x}}dx=\sqrt{2\pi }\mathrm{S}\sqrt{\frac{2x}{\pi }}+C$$$$$$$$(\ast )\mathrm{let}u=\sqrt{\frac{2x}{\pi }}\Rightarrow dx=\sqrt{2\pi x}$$

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