$\int(\frac{1}{\sqrt{x}})\sin(\frac{1}{x}) dx$

Can this be expressed in terms of elementary functions ?

  • I revised your question. Double check to see if it is what you meant to ask.

    • Gubiv Gubiv

      Yes thank you very much

1 Answer

No, it can not be written in terms of elementary functions. Indeed

\[\int \frac{1}{\sqrt{x}}\sin(\frac{1}{x})dx=2\sin\left(\dfrac{1}{x}\right)\sqrt{x}-2^\frac{3}{2}\sqrt{{\pi}}\operatorname{G}\left(\dfrac{\sqrt{2}}{\sqrt{{\pi}}\sqrt{x}}\right)+C,\]
where $G$ is the Fresnel integral 
\[G(u)=\int \cos (\frac{\pi u^2}{2})du.\]

Use the following website for more details: https://www.integral-calculator.com/ 

For more information about Fresnel integrals see  https://en.wikipedia.org/wiki/Fresnel_integral

Erdos Erdos
  • Gubiv Gubiv

    Oh okay thanks for the answer. But would I be able to come to this solution on paper within reasonable time, or is it something that requires the aid of an advanced calculator

    • Erdos Erdos

      Yes, you just need to do an integration by parts and some basic change of variables. I suggest you to use the first website I mentioned above. It will show you the computations.

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