$\int(\frac{1}{\sqrt{x}})\sin(\frac{1}{x}) dx$
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
-
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
-
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.
-
- 1 Answer
- 322 views
- Pro Bono
Related Questions
- Calculus - stationary points, Taylor's series, double integrals..
- Rewrite $\int_{\sqrt2}^{2\sqrt2} \int_{-\pi/2}^{-\pi/4} r^2cos(\theta)d\theta dr$ in cartesian coordinates (x,y)
- Line Integral
- Evaluate the integral $\int_{-\infty}^{+\infty}e^{-x^2}dx$
- Please answer the attached question about Riemann integrals
- Evaluate $\int \sin x \sqrt{1+\cos x} dx$
- Finding absolute and relative extrema given an equation.
- Compound Interest question
I revised your question. Double check to see if it is what you meant to ask.
Yes thank you very much