Linked Questions
32 questions linked to/from Evaluating $\int_0^\infty \sin x^2\, dx$ with real methods?
16
votes
4
answers
33k
views
Some way to integrate $\sin(x^2)$? [duplicate]
Because the straight forward approach involves Fresnel integrals I thought about a different approach of taking the imaginary part of
$\int_{-\infty}^{\infty}\exp{(ix^2)} \, dx $ but have no idea how ...
- 1,137
3
votes
1
answer
528
views
Evaluate $\int_{0}^{\infty} \cos(x^2)dx $ [duplicate]
Prove that the above integral is equal to $\frac{\sqrt{2\pi}}{2}$
I have already tried expanding using $\cos$ identity and also taking Laplace for it.
I am getting nowhere with this.
- 41
2
votes
1
answer
346
views
How to calculate $\int_{0}^{+\infty} {\sin {x^2}\mathrm{d}x}$? [duplicate]
Possible Duplicate:
Evaluating $\int_0^\infty \sin x^2\, dx$ with real methods?
How to calculate $\displaystyle\int_{0}^{+\infty} {\sin {x^2}\mathrm{d}x}$ ?
- 166
216
votes
20
answers
152k
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Evaluation of Gaussian integral $\int_{0}^{\infty} \mathrm{e}^{-x^2} dx$
How to prove
$$\int_{0}^{\infty} \mathrm{e}^{-x^2}\, dx = \frac{\sqrt \pi}{2}$$
- 7,938
43
votes
7
answers
33k
views
Prove: $\int_0^\infty \sin (x^2) \, dx$ converges.
$\sin x^2$ does not converge as $x \to \infty$, yet its integral from $0$ to $\infty$ does.
I'm trying to understand why and would like some help in working towards a formal proof.
- 7,020
22
votes
8
answers
8k
views
Proof of $\int_0^\infty \frac{\sin x}{\sqrt{x}}dx=\sqrt{\frac{\pi}{2}}$
Numerically it seems to be true that
$$
\int_0^\infty \frac{\sin x}{\sqrt{x}}dx=\sqrt{\frac{\pi}{2}}.
$$
Any ideas how to prove this?
- 221
41
votes
5
answers
3k
views
Proof only by transformation that : $ \int_0^\infty \cos(x^2) dx = \int_0^\infty \sin(x^2) dx $
This was a question in our exam and I did not know which change of variables or trick to apply
How to show by inspection ( change of variables or whatever trick ) that
$$ \int_0^\infty \cos(x^2) dx ...
- 23.5k
37
votes
5
answers
4k
views
Generalizing the trick for integrating $\int_{-\infty}^\infty e^{-x^2}\mathrm dx$?
There is a well-known trick for integrating $\int_{-\infty}^\infty e^{-x^2}\mathrm dx$, which is to write it as $\sqrt{\int_{-\infty}^\infty e^{-x^2}\mathrm dx\int_{-\infty}^\infty e^{-y^2}\mathrm dy}$...
user13618
33
votes
5
answers
2k
views
tough integral involving $\sin(x^2)$ and $\sinh^2 (x)$
I ran across this integral I get no where with. Can someone suggest a method of attack?.
$$\int_0^{\infty}\frac{\sin(\pi x^2)}{\sinh^2 (\pi x)}\mathrm dx=\frac{2-\sqrt{2}}{4}$$
I tried series, ...
- 13.9k
8
votes
6
answers
7k
views
Compute the value of Fresnel's Integral
So, my teacher wants us to compute the value of the Fresnel integral:
$$\int_0^\infty\cos(x^2)dx=\sqrt{\frac{\pi}{8}}$$
The problem is that we cannot use complex analysis to prove that and we should ...
- 105
13
votes
4
answers
6k
views
Do integrable functions vanish at infinity? [duplicate]
If $f$ is a real-valued function that is integrable over $\mathbb{R}$, does it imply that
$$f(x) \to 0 \text{ as } |x| \to \infty? $$
When I consider, for simplicity, positive function $f$ which is ...
- 761
10
votes
3
answers
1k
views
Meaning and example(s) of Qiaochu's quote.
I happen to come across this page http://math.uchicago.edu/~chonoles/quotations.html which contains some beautiful quotes by various mathematicians and I came across Qiaochu's quote as claimed by the ...
15
votes
3
answers
2k
views
Is Gaussian integral the only one that can be easily solved by this double integral trick?
For a lot of people the favorite way of solving Gaussian integral $I=\int^{\infty}_{-\infty} e^{-x^2} dx$ is to find $I^2$ in polar coordinates and then take a root.
The trick may be useful in this ...
- 31.3k
15
votes
2
answers
474
views
Evaluating $ \int^{\infty}_{-\infty}\sin\left({\pi}^{4}x^{2}+\frac{1}{x^2}\right) dx$
$$\int^{\infty}_{-\infty}\sin\left({\pi}^{4}x^{2}+\frac{1}{x^2}\right) dx$$
This is a problem from the Pi Mu Epsilon Journal, and I'm having great trouble answering it. I've tried some substitutions ...
- 153
8
votes
2
answers
2k
views
Trig Fresnel Integral
$$\int_{0}^{\infty }\sin(x^{2})dx$$
I'm confused with this integral because the square is on the x, not the whole function. How can I integrate it? Thank you.
I have not done complex analysis (only ...
- 1,542