Questions on the Fresnel integrals.

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33
votes
3answers
830 views

An integral involving Fresnel integrals $\int_0^\infty \left(\left(2\ S(x)-1\right)^2+\left(2\ C(x)-1\right)^2\right)^2 x\ \mathrm dx,$

I need to calculate the following integral: $$\int_0^\infty \left(\left(2\ S(x)-1\right)^2+\left(2\ C(x)-1\right)^2\right)^2 x\ \mathrm dx,$$ where $$S(x)=\int_0^x\sin\frac{\pi z^2}{2}\mathrm dz,$$ ...
11
votes
1answer
183 views

Need help with $\int_0^\infty\left(\pi\,x+\frac{S(x)\cos\frac{\pi x^2}2-C(x)\sin\frac{\pi x^2}2}{S(x)^2+C(x)^2}\right)dx$

Let $$I=\int_0^\infty\left(\pi\,x+\frac{S(x)\cos\frac{\pi x^2}2-C(x)\sin\frac{\pi x^2}2}{S(x)^2+C(x)^2}\right)dx,\tag1$$ where $$S(x)=-\frac12+\int_0^x\sin\frac{\pi t^2}2dt,\tag2$$ ...
5
votes
1answer
140 views

Definite integral involving Fresnel integrals

I am seeking to evaluate $\int_0^{\infty} f(x)/x^2 \, dx$ with $f(x)=1-\sqrt{\pi/6} \left(\cos (x) C\left(\sqrt{\frac{6 x}{\pi }} \right)+S\left(\sqrt{\frac{6 x}{\pi }} \right) \sin ...
4
votes
2answers
565 views

Are Complex Substitutions Legal in Integration?

This question has been irritating me for awhile so I thought I'd ask here. Are complex substitutions in integration okay? Can the following substitution used to evaluate the Fresnel integrals: ...
4
votes
1answer
628 views

Connect two curves with Euler spiral segment

Image of situation: http://upload.wikimedia.org/wikipedia/commons/5/54/Easement_curve.svg Let's say we have a straight line (blue) and a circular arc (green). My goal is to connect these two curves ...
1
vote
1answer
188 views

Fresnel Integrals via Differentiation under the Integral Sign

I've been trying to compute $\int_{-\infty}^{\infty}sin( x^2)dx$ via the feynman method with no luck. I was able to compute the Gaussian integral. The trick failed for fresnel integrals. Any ...
1
vote
0answers
33 views

Compute Erf(z) using Fresnel integrals

I have to compute $\operatorname{erf}(z)$ using the Fresnel integrals. I have the relation: $$C(z)+iS(z)=\frac{1+i}{2}\operatorname{erf}\left[ \frac{\sqrt{\pi}}{2}(1-i)z \right].$$ But ...
0
votes
2answers
62 views

Evaluating $\int_0^{\frac{\pi}2}\frac{\sin 2x}{\sqrt{x}}\,dx$

$$\int_0^{\frac{\pi}2}\frac{\sin 2x}{\sqrt{x}}\,dx$$ How to solve this trigonometric integral? I can't find any solutions. Some books suggest to use Fresnel integral. I would be grateful if you could ...
0
votes
1answer
82 views

IB Math question: Fresnel integral?

P moves along x-axis such that its velocity, v, at time t is given by $v=\cos(t^2)$. Find the time at which the total distance travelled by P is 1. (all in meters, meters/sec). So the total distance ...
0
votes
1answer
66 views

how to prove an identity related to $\int_0^\infty\sin(x^{1+a})dx$?

i have made some experiments in maple evaluating the integral $$\int_0^\infty\sin(x^{1+a})dx$$ and the computer give me the following result ...
0
votes
1answer
85 views

Evaluating $\int_0^1 \! C(x) \, \mathrm dx$ through integration by parts

$$ \int_0^1 \! C(x) \, \mathrm{d} x. $$ where $C(x) = \int_0^x \cos(t^2) \, \mathrm{d} t$. I am really not quite sure how to go about this one, especially given that it needs to be calculated ...
0
votes
0answers
56 views

How to solve $\int_0^\infty \sin(x^2)\,{\rm d}x$

I was trying to solve this Fresnel integral using Feynman integration(differentiation under the integral sign) $$\int_{-\infty}^\infty \sin(x^2)\,{\rm d}x$$ So we define, $$F(b)= \int_0^\infty ...