How to solve this definite integral; one with a function in an exponential.

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My integral table has this definite integral:



$$int_-infty^infty e^-big(a x^2 + b x + cbig) dx = sqrtfracpiae^fracb^2-4ac4a$$



I'd like to solve this similar definite integral:



$$int_-infty^infty e^-ibig(a x^2 + b x + c(x)big) dx~~=~~~~?$$



Where $x$ is real valued, $a$ and $b$ are constants, and $c(x)$ is an arbitrary function of $x$, and $i=sqrt-1$. The big difference between my equation and the one I found on the table is that, $c(x)ne c$.



What is the solution to this? Or, if it isn't easy to solve, what steps should I take? There are a few ways I can limit $c(x)$; for example, it could be Gaussian white noise. However, I'd prefer to solve the general case. Or is there an approximation?




integration definite-integrals




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  • Use Fresnel integral as a reference.
    – Gustave
    Jul 30 at 23:45






  • 3




    I may be wrong, but I think you have to be more specific about the nature of $c(x)$ to answer this question
    – angryavian
    Jul 30 at 23:46










  • @MarkViola op has a function of x in $c(x)$ that is unknown
    – Isham
    Jul 30 at 23:53






  • 3




    if $c(x)$ can be any function of x then it's just like trying to solve the general case $$int_-infty^infty e^c(x)dx$$
    – Isham
    Jul 30 at 23:57







  • 2




    Right. There is no way to help without specification of $c(x)$. Just imagine that $c(x) = sin (1/x)$ or $1$ or $tan (x^2)$ or ...
    – David G. Stork
    Jul 31 at 1:23















up vote
0
down vote

favorite












My integral table has this definite integral:



$$int_-infty^infty e^-big(a x^2 + b x + cbig) dx = sqrtfracpiae^fracb^2-4ac4a$$



I'd like to solve this similar definite integral:



$$int_-infty^infty e^-ibig(a x^2 + b x + c(x)big) dx~~=~~~~?$$



Where $x$ is real valued, $a$ and $b$ are constants, and $c(x)$ is an arbitrary function of $x$, and $i=sqrt-1$. The big difference between my equation and the one I found on the table is that, $c(x)ne c$.



What is the solution to this? Or, if it isn't easy to solve, what steps should I take? There are a few ways I can limit $c(x)$; for example, it could be Gaussian white noise. However, I'd prefer to solve the general case. Or is there an approximation?




integration definite-integrals




share|cite|improve this question



















  • Use Fresnel integral as a reference.
    – Gustave
    Jul 30 at 23:45






  • 3




    I may be wrong, but I think you have to be more specific about the nature of $c(x)$ to answer this question
    – angryavian
    Jul 30 at 23:46










  • @MarkViola op has a function of x in $c(x)$ that is unknown
    – Isham
    Jul 30 at 23:53






  • 3




    if $c(x)$ can be any function of x then it's just like trying to solve the general case $$int_-infty^infty e^c(x)dx$$
    – Isham
    Jul 30 at 23:57







  • 2




    Right. There is no way to help without specification of $c(x)$. Just imagine that $c(x) = sin (1/x)$ or $1$ or $tan (x^2)$ or ...
    – David G. Stork
    Jul 31 at 1:23













up vote
0
down vote

favorite









up vote
0
down vote

favorite











My integral table has this definite integral:



$$int_-infty^infty e^-big(a x^2 + b x + cbig) dx = sqrtfracpiae^fracb^2-4ac4a$$



I'd like to solve this similar definite integral:



$$int_-infty^infty e^-ibig(a x^2 + b x + c(x)big) dx~~=~~~~?$$



Where $x$ is real valued, $a$ and $b$ are constants, and $c(x)$ is an arbitrary function of $x$, and $i=sqrt-1$. The big difference between my equation and the one I found on the table is that, $c(x)ne c$.



What is the solution to this? Or, if it isn't easy to solve, what steps should I take? There are a few ways I can limit $c(x)$; for example, it could be Gaussian white noise. However, I'd prefer to solve the general case. Or is there an approximation?




integration definite-integrals




share|cite|improve this question











My integral table has this definite integral:



$$int_-infty^infty e^-big(a x^2 + b x + cbig) dx = sqrtfracpiae^fracb^2-4ac4a$$



I'd like to solve this similar definite integral:



$$int_-infty^infty e^-ibig(a x^2 + b x + c(x)big) dx~~=~~~~?$$



Where $x$ is real valued, $a$ and $b$ are constants, and $c(x)$ is an arbitrary function of $x$, and $i=sqrt-1$. The big difference between my equation and the one I found on the table is that, $c(x)ne c$.



What is the solution to this? Or, if it isn't easy to solve, what steps should I take? There are a few ways I can limit $c(x)$; for example, it could be Gaussian white noise. However, I'd prefer to solve the general case. Or is there an approximation?





integration definite-integrals





share|cite|improve this question










share|cite|improve this question




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asked Jul 30 at 23:41









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  • Use Fresnel integral as a reference.
    – Gustave
    Jul 30 at 23:45






  • 3




    I may be wrong, but I think you have to be more specific about the nature of $c(x)$ to answer this question
    – angryavian
    Jul 30 at 23:46










  • @MarkViola op has a function of x in $c(x)$ that is unknown
    – Isham
    Jul 30 at 23:53






  • 3




    if $c(x)$ can be any function of x then it's just like trying to solve the general case $$int_-infty^infty e^c(x)dx$$
    – Isham
    Jul 30 at 23:57







  • 2




    Right. There is no way to help without specification of $c(x)$. Just imagine that $c(x) = sin (1/x)$ or $1$ or $tan (x^2)$ or ...
    – David G. Stork
    Jul 31 at 1:23

















  • Use Fresnel integral as a reference.
    – Gustave
    Jul 30 at 23:45






  • 3




    I may be wrong, but I think you have to be more specific about the nature of $c(x)$ to answer this question
    – angryavian
    Jul 30 at 23:46










  • @MarkViola op has a function of x in $c(x)$ that is unknown
    – Isham
    Jul 30 at 23:53






  • 3




    if $c(x)$ can be any function of x then it's just like trying to solve the general case $$int_-infty^infty e^c(x)dx$$
    – Isham
    Jul 30 at 23:57







  • 2




    Right. There is no way to help without specification of $c(x)$. Just imagine that $c(x) = sin (1/x)$ or $1$ or $tan (x^2)$ or ...
    – David G. Stork
    Jul 31 at 1:23
















Use Fresnel integral as a reference.
– Gustave
Jul 30 at 23:45




Use Fresnel integral as a reference.
– Gustave
Jul 30 at 23:45




3




3




I may be wrong, but I think you have to be more specific about the nature of $c(x)$ to answer this question
– angryavian
Jul 30 at 23:46




I may be wrong, but I think you have to be more specific about the nature of $c(x)$ to answer this question
– angryavian
Jul 30 at 23:46












@MarkViola op has a function of x in $c(x)$ that is unknown
– Isham
Jul 30 at 23:53




@MarkViola op has a function of x in $c(x)$ that is unknown
– Isham
Jul 30 at 23:53




3




3




if $c(x)$ can be any function of x then it's just like trying to solve the general case $$int_-infty^infty e^c(x)dx$$
– Isham
Jul 30 at 23:57





if $c(x)$ can be any function of x then it's just like trying to solve the general case $$int_-infty^infty e^c(x)dx$$
– Isham
Jul 30 at 23:57





2




2




Right. There is no way to help without specification of $c(x)$. Just imagine that $c(x) = sin (1/x)$ or $1$ or $tan (x^2)$ or ...
– David G. Stork
Jul 31 at 1:23





Right. There is no way to help without specification of $c(x)$. Just imagine that $c(x) = sin (1/x)$ or $1$ or $tan (x^2)$ or ...
– David G. Stork
Jul 31 at 1:23
















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