$\int(\frac{1}{\sqrt{x}})\sin(\frac{1}{x}) dx$
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
- 399 views
- Pro Bono
Related Questions
- Find amplitude-frequency characteristic of a discrete finite signal using Z-transform
- Differentiate $\int_{\sin x}^{\ln x} e^{\tan t}dt$
- Find $\lim_{x \rightarrow} x^2 \sin(1/x) $. Cite theorems used and show all work
- Some final thoughts and closing out this question
- Find the area under the graph of $y=\sin x$ between $x=0$ and $x=\pi$.
- Minimizing the cost of building a box
- Recursive square root sequence
- Differentiate $f(x)=\int_{\sqrt{x}}^{\arcsin x} \ln\theta d \theta$
I revised your question. Double check to see if it is what you meant to ask.
Yes thank you very much