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How can I calculate Fresnel integral via double improper integral?

Show that $intlimits_{-infty}^{infty} ! mathbb{e}^{-x^2} , mathrm{d}x$ = $sqrt{pi}$.How to calculate the improper integral $int_{0}^{infty}Big(frac{x}{mathrm{e}^x-mathrm{e}^{-x}}-frac{1}{2}Big)frac{mathrm{d}x}{x^2}$Does the improper integral $int_{0}^{infty}sin(x^2);mathrm dx$ converge?Find a value of a strange improper integral.Improper integral of trigonometric functionAn almost Fresnel integralCalculate the double integral over a region $int_{D}(x^2+y^2)dxdy$Strange Limit in Proof of the Fresnel Integralalternative definition of improper Riemann integralProof that improper integral $int_{0}^{infty} e^{-x}(1+sin(x^2)) dx$ exists

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So, there are many ways to calculate the integral $int_{-infty}^{+infty} sin(x^2),mathrm{d}x$. For example, you can use differentiation under the integral sign or you can use complex numbers. My teacher asked me to find the value of this Fresnel integral using double improper integral. Perhaps one may use $iintlimits_mathbb{R^2} sin(x^2 + y^2),mathrm{d}x,mathrm{d}y$? But I don’t know yet how this works here.

calculus improper-integrals multiple-integral fresnel-integrals

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edited Mar 14 at 21:32

Brian

984116

asked Mar 14 at 21:29

John D.John D.

12

$endgroup$

add a comment | 

0

$begingroup$

So, there are many ways to calculate the integral $int_{-infty}^{+infty} sin(x^2),mathrm{d}x$. For example, you can use differentiation under the integral sign or you can use complex numbers. My teacher asked me to find the value of this Fresnel integral using double improper integral. Perhaps one may use $iintlimits_mathbb{R^2} sin(x^2 + y^2),mathrm{d}x,mathrm{d}y$? But I don’t know yet how this works here.

calculus improper-integrals multiple-integral fresnel-integrals

share|cite|improve this question

edited Mar 14 at 21:32

Brian

984116

asked Mar 14 at 21:29

John D.John D.

12

$endgroup$

add a comment | 

$begingroup$

So, there are many ways to calculate the integral $int_{-infty}^{+infty} sin(x^2),mathrm{d}x$. For example, you can use differentiation under the integral sign or you can use complex numbers. My teacher asked me to find the value of this Fresnel integral using double improper integral. Perhaps one may use $iintlimits_mathbb{R^2} sin(x^2 + y^2),mathrm{d}x,mathrm{d}y$? But I don’t know yet how this works here.

calculus improper-integrals multiple-integral fresnel-integrals

share|cite|improve this question

edited Mar 14 at 21:32

Brian

984116

asked Mar 14 at 21:29

John D.John D.

12

$endgroup$

share|cite|improve this question

edited Mar 14 at 21:32

Brian

984116

asked Mar 14 at 21:29

John D.John D.

12

share|cite|improve this question

edited Mar 14 at 21:32

Brian

984116

asked Mar 14 at 21:29

John D.John D.

12

edited Mar 14 at 21:32

Brian

984116

edited Mar 14 at 21:32

Brian

984116

asked Mar 14 at 21:29

John D.John D.

12

asked Mar 14 at 21:29

John D.John D.

12