# Tagged Questions

Questions about the evaluation of specific definite integrals.

388 views

295 views

### Closed form for this integral $\int_{0}^{\infty}\frac{dx}{\sqrt{x}}\, e^{-x^{2}-\frac{b^{2}}{x}}$

How would you evaluate this integral? $$\Large\int_{0}^{\infty}\frac{dx}{\sqrt{x}}\, e^{-x^{2}-\frac{b^{2}}{x}}$$ It reminds me of the form of a modified Bessel function of ...
238 views

### Proof of Cauchy's Beta Integral $\int_{-\infty}^\infty \frac{dt}{(1+it)^x(1-it)^y}$

The Cauchy's Beta Integral is given by $$\int_{-\infty}^\infty \frac{dt}{(1+it)^x(1-it)^y}=\frac{\pi 2^{2-x-y}\Gamma(x+y-1)}{\Gamma(x)\Gamma(y)}$$ I would like to know how it is proved.
187 views

45 views

### Integrals 3.384 from Gradshteyn and Ryzhik

I'm interested in understanding the computation of $$\int_{-\infty}^\infty\frac{e^{-ip x}}{(1 + ix)^{2u}(1-ix)^{2v}}\mathrm{d}x,$$ which is evaluated in 3.384.9 of Gradshteyn and Rhysik for ...
57 views

### Is there a closed form for this sum?

While generalizing the previous result, I conjectured that the series expansion of \begin{align*} \int_{0}^{\frac{\pi}{2}} \arctan \left( \frac{2x \sin\theta}{1-x^{2}} \right) \arctan \left( \frac{2y ...
89 views

### Need help with $\int_0^2\frac{1}{2+\sqrt{3\,e^x+3\,e^{-x}-2}}dx$

Could you please help me to solve this integration problem? $$\int_0^2\frac{1}{2+\sqrt{3\,e^x+3\,e^{-x}-2}}dx$$ Its approximate numeric value is $0.419197813818367...$, but I could not find an exact ...
67 views

### A closed form for $\int_0^\infty\left(\frac{2^{-x}-3^{-x}}x\right)^adx,\ a\notin\mathbb{Z}^+$

Let $$I(a)=\int_0^\infty\left(\frac{2^{-x}-3^{-x}}x\right)^adx.$$ $I(a)$ has closed form representations for all $a\in\mathbb{Z}^+$. Is there any algebraic (or at least period) ...
38 views

63 views

49 views

### How to integrate the bump functions,i.e,$\int_a^{b}e^{-\frac{1}{x-a}+\frac{1}{x-b}}dx$,where $a<b$.

Since $$\lim_{x\to{a}}e^{-\frac{1}{x-a}+\frac{1}{x-b}}=\lim_{x\to{b}}e^{-\frac{1}{x-a}+\frac{1}{x-b}}=0,$$ $e^{-\frac{1}{x-a}+\frac{1}{x-b}}$ is continuous on the interval $[a,b]$ (taking $0$ if ...
50 views

31 views

### fancy about some properties of kernel functions at infinity

Consider the two common types of kernel functions $\sum\limits_{t=a}^bf(t)K(x,t)$ and $\int_a^bf(t)K(x,t)~dt$ , prove whether the following properties are correct or not: $1.$ If $K(x,t)$ is bounded ...
112 views

### Simplify the integral with error function

$\newcommand{\erf}{\operatorname{erf}}$ I have the following integral and I need to simplify the solution. I have written first two steps. I don't know what is the value of $$\erf(\infty)$$ I ...
136 views

68 views

### How do you set up the integral in spherical coordinates in the following problem?

Find the volume bounded by the surface $z = x^2 + y^2$ and $x^2+y^2 = 1$ in the first quadrant. The answer is $\pi/8$ using rectangular and cylindrical coordinates and that is the correct answer, but ...
51 views

### Minimizing the integral

I need help in the following question. Let $$J(u)=\int_0^1\left[u_x^2+4\frac{u^2}{x^2}\right]xdx,$$ where $u(x)$ is a smooth function defined on $[0,1]$ satisfyig $u(0)=0$ and $u(1)=1$. Which one of ...
382 views

### Integral with Legendre polynomials

What is $$\int_{\frac{\pi}{2}}^{\pi} P_l(\cos(\theta))P_{l'}(\cos(\theta)) \sin(\theta) d\theta$$ in general? Of course, $P_l$ is the l-th Legendre polynomial. Obvisiouly it is for even/even and ...
55 views

### Calculating a definite integral

$$\int_{0}^{T}\frac{e^{-\alpha x}-e^{-\beta x}}{1-e^{-\gamma x}}dx$$ Mathematica gives an answer based on the Hypergeometric Function. Could there be a closed form?
125 views

### Maxima of $\int_0^a f(x)^2 \ dx$ and $\int_0^a xf(x)^2 \ dx$

Let $f:[0,1]\rightarrow[0,\infty)$ be an increasing function, $a \in (0,1)$, and $\displaystyle \int_0^1 f(x) \ dx =1$. What are the maxima of $$i)\int_0^a f(x)^2 \ dx$$ $$ii)\int_0^a xf(x)^2 \ dx$$ ...
59 views