Articles of integration

Solving an ODE from a PDE

Solve this differential equation: $$z^2\frac{dR}{R}+\frac{z\,dz}{1+z^2}=\frac{dS}{S}\tag1$$ I tried separating the variables but could not. Also tried to show that LHS is of the form of the differential of a product of functions but could not. Originally this is coming from a PDE $$z\frac{\partial n}{\partial x}dx+\frac{n\,dz}{1+z^2}=\frac{1}{z}\cdot \frac{\partial n}{\partial y}dy\tag2$$ where $z=\frac{dy}{dx}$ and $n=n(x,y)$ and $y=y(x)$. The first […]

Numerical value of $\int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy $

Could somebody give me a numerical value for this integral? $$I = \int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy $$

How to solve the following integral?

I am trying to evaluate the following integral by parts, but no answers so far.$$\int_{0}^{\infty}\left[\frac{x}{(1+ax)(1+x)}\right]^{K}dx,$$ where $K\in\mathbb{N}$ and $a>0$ is a constant. Can anyone help me?

Integral related to a geometry problem

In the question Geometry problem involving infinite number of circles I showed that the answer could be obtained by the sum $$ \sum_{k=0}^{\infty}\int_{B_{k}} {4 \over \,\left\vert\,1 + \left(\,x + y{\rm i}\,\right)\,\right\vert^{\,4}\,} \,{\rm d}x\,{\rm d}y\,, $$ where $ B_k = \left\{\, z \in \mathbb{C}:\ \left\vert\,z – \left[\frac{1}{2} + \left(k + {1 \over 2}\right) {\rm i}\right]\,\right\vert\ \leq\ […]

Do we have $\lim_{h\to 0^+}\int_0^\infty|f(t+h)-f(t)|dt=0$?

Let $f:\mathbb{R^+}\to\mathbb{R}$ be an integrable function ($f\in L^1(\mathbb{R}^+,\mathbb{R})$). Do we have $$\lim_{h\to 0^+}\int_0^\infty|f(t+h)-f(t)|dt=0$$ ? How can we prove it ?

Endpoint-average inequality for a line segment in a normed space

Let $X$ be a normed vector space over $\mathbb R$. What is the smallest universal constant $C>0$ such that the inequality $$\|x\|\le C\int_0^1 \|x+tv\|\,dt\tag{1}$$ holds for all $x,v\in X$? Geometrically, (1) means that the norm of an endpoint of a line segment can be majorized by the average of the norm over said line segment. […]

Simple Integral Involving the Square of the Elliptic Integral

I have, $$ \int uE^{2}\left(u\right)du $$ where $E$ is the complete elliptic integral of the second kind: $$ E\left(k\right)=\int_{0}^{\frac{\pi}{2}}d\theta\sqrt{1-k^{2}\sin^{2}\left(\theta\right)} $$ I’ve tried integrating this by looking for something that differentiates into $E^2(u)$, though can’t seem to find anything. Any help?

A closed form for the integral $\int_0^1\frac{1}{\sqrt{y^3(1-y)}}\exp\left(\frac{i A}{y}+\frac{i B}{1-y}\right)dy$

Yesterday, during reviewing my old lecture notes on advanced quantum mechanics, i stumbeled over the following integral identity, which seems, on a first glance, too nice to be true $$ I_{A,B}=\int_0^1\frac{1}{\sqrt{y^3(1-y)}}\exp\left(\frac{i A}{y}+\frac{i B}{1-y}\right)dy=\sqrt{\frac{i\pi}{B }}e^{i(\sqrt{A}+\sqrt{B})^2} $$ with $A,B>0$ After working on it for a few hours i came up with a solution, which i think is […]

Topology on the space of universally integrable functions

Let $X$ be a compact space. Let us call a function $f:X\to {\mathbb C}$ universally integrable if it is integrable with respect to each regular Borel measure $\mu$ on $X$ (i.e. a positive functional on ${\mathcal C}(X)$, according to Riesz–Markov–Kakutani representation theorem). Let us denote by ${\mathcal U}(X)$ the space of all universally integrable functions […]

Definite integral of a product of normal pdf and cdf

Denote the pdf of the standard normal distribution as $\phi(x)$ and cdf as $\Phi(x)$. Does anyone know how to calculate $\int_{-\infty}^y \phi(x)\Phi(\frac{x−b}{a})dx$? Notice that this question is similar to an existing one, the only difference being that I’m computing the integral over $(-\infty, y)$ for some real $y$, rather than over the entire real […]