Linked Questions

7
votes
3answers
213 views

Solving the Definite Integral $\int_0^{\infty} \frac{1}{t^{\frac{3}{2}}} e^{-\frac{a}{t}} \, \mathrm{erf}(\sqrt{t})\, \mathrm{d}t$

I would like to solve the following integral $$\int_0^{\infty} \frac{1}{t^{\frac{3}{2}}} e^{-\frac{a}{t}} \, \mathrm{erf}(\sqrt{t})\, \mathrm{d}t$$ with Re$(a)>0$ and erf the error function. Is ...
4
votes
2answers
1k views

hard integral problems to solve

I'm practicing harder integration using techniques of solving with special functions I have difficulties with these two hard integrals; don't even know how to start, $$\int_0 ^\infty x^p e^{-\frac{\...
7
votes
3answers
404 views

What is the Fourier transform of $e^{2 \pi i / x}$?

The Fourier transform of $e^{2 \pi i / x}$ makes sense as a distribution, I believe. Does it have a nice expression in terms of functions and well-known distributions (e.g. Dirac delta)?
7
votes
2answers
225 views

A Sine integral: problem I

Is it possible to demonstrate a solution for the integral \begin{align} \int_{0}^{\infty} x^{n} \, \sin\left( a x^{2} + \frac{b}{x^{2}} \right) \, dx \end{align}
0
votes
2answers
239 views

More difficult Integral

This is sort of follow up to my previous question (less difficult integral) here. How do I find $$\large\int_{0}^{\infty}x^ne^{-\left(ax+\frac{b}{x}\right)}dx$$ where $a$ and $b$ are positive reals, $...
4
votes
3answers
185 views

Does anyone know how to calculate the following integral?

Consider the function (coming from a joint probability density): $$ f(x,y) = \frac{1}{y}e^{-y-\frac{x}{y}}. $$ I want to evaluate the definite integral (marginal): $$ F(x) = \int_0^\infty f(x,y)\,dy. ...
4
votes
2answers
201 views

How to compute $\int_0^{\infty}\sqrt x \exp\left(-x-\frac{1}{x}\right) \, dx$?

How to compute this integral? : $$\int_0^{\infty}\sqrt x \exp\left(-x-\frac{1}{x}\right) \, dx$$ Wolframalpha gives the answer $\dfrac{3\sqrt{\pi}}{2e^2}$, but how to compute this?
3
votes
2answers
121 views

Integral of $1/x^2 \exp(-ax+b-c/x)dx$

I am interested in simplifying the integral $$\int \frac{1}{x^2}\exp(-ax+b-\frac{c}{x})dx$$ with $a, b, c \in \mathbb{R}$. Do you have any idea? With $a=0$ or $c=0$ I know the solutions but what about ...
1
vote
2answers
97 views

An improper integrals related to probability, $\int_0^\infty\frac1y \exp(\frac{-x_0}y-y)\,dy$

How can I calculate the integral $$\int_0^\infty{\frac1y e^{\frac{-x_0}y-y}}dy$$ in terms of well-known constants and functions? I used some fundamental techniques of integration but got nothing.
3
votes
1answer
116 views

The closed form of $\int^\infty_{B}e^{-(x+\frac{A}{x})}\,dx$, where $A>0$, $B>0$.

What tools, ways would you propose for getting the closed form of this integral? $$\int^\infty_{B}e^{-\left(x+\frac{A}{x}\right)}\,dx,$$ where $A>0$, $B>0$. When $B=0$, from Table of Integrals,...
2
votes
3answers
68 views

Query related to Eq. 3.471.9 of Book of Gradeshteyn (Integration tables series and products)

Equation no. 3.471.9 of Integral series and products (By Gradeshteyn) is written below $$\int_0^{\infty}x^{v-1}e^{-\frac{\beta}{x}-\gamma x}dx=2\left(\frac{\beta}{\gamma}\right)^{\frac{v}{2}}K_{v}(2\...
1
vote
1answer
96 views

How to rewrite this integral $I = \int e^{ - \left( {ax + \frac{b}{x}} \right)} dx$ as non-elementary function?

Is it possible to rewrite or evaluate this integral $I = \int\limits_1^p e^{ - \left( {ax + \frac{b}{x}} \right)} dx$ where $a,b,p > 0$ as some known non-elementary function (For example $\...