Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of ...

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476 views

Taylor Series of Integral

I'm trying to come up with the Taylor expansion of an integral expression. For simplicity, consider the toy integral $$ ...
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122 views

An interesting integral

How to integrate: $$ \int \frac{x}{\sqrt{x^4+10x^2-96x-71}}.$$ I read about this problem on the Wikipedia Risch algorithm page, they gave an answer but I am at a loss how they got it....
3
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133 views

Integrate: $\int\limits_0^\infty{\frac{x^{n-2}}{b\left(1+ ~a x^{\frac{n-1}{n-2}}\right)} \sin{(x b)}~ dx}$

I am trying to solve the integral: $\int\limits_0^\infty{\frac{x^{n-2}}{b\left(1+ ~a x^{\frac{n-1}{n-2}}\right)} \sin{(x b)}~ dx}$ where $x$ is real and $a, b, n$ are positive real constants. I ...
3
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136 views

Attempted proof of the first part of the Fundamental Theorem of Calculus

So I was trying to prove the first part of the Fundamental Theorem of Calculus in a different way using the Riemann sum definition of the definite integral, rather than the way it was presented in my ...
3
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101 views

Transforming a Riemann-Stieltjes integral related to the factorial

I have been able to show that $$\log n! = \int_{1 + \epsilon}^n \log x \, d\lfloor x \rfloor$$ but I have not been successful trying to transform this Riemann-Stieltjes integral to an ordinary ...
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178 views

integral involving upper incomplete gamma function

I trying to evaluate the following integral $$\int_0^\infty \dfrac { x^{m-1} \Gamma(A,\mathcal B x^q)} {\left[1+(\eta x)^n\right]^p} \,\mathrm dx$$ where the integration is w.r.t. $x$, and the ...
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99 views

Solving exponential integral

Any idea how to solve this integral, I tried the integration by parts, and it made the things even more difficult. The substitution didn't work either. Here is the integral: $\displaystyle \int ...
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38 views

Is this a decomposition of the same function?

Let's say we have some integral, such that for a particular function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ $$\int_{\mathbb{R}^{n-m}} \int_{\mathbb{R}^m}f^+ - ...
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81 views

Determine the behavior of a function defined by an integral

Suppose we have a function defined by $$\varphi(s)=\int_{-\infty}^\infty f(x,s)\,dx$$ defined for $s\in S\subseteq \mathbb{R}$. Suppose we know that it blows up at $a\in \partial S$, and we want to ...
3
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102 views

Equivalent definitions of Fourier transform of a measure

For me the fourier transform of a measure $\mu\in\mathcal{S}'(\mathbb{R})$ is defined by $\hat{\mu}(\varphi)=\mu(\hat{\varphi})$ where $\varphi\in\mathcal{S}(\mathbb{R})$. My question is: if one has ...
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97 views

Two properties about Bessel function

Let $J_\nu(x)$ be the Bessel function of the first kind. $\int_0^\infty J_\nu(x)dx=1 , (Re(\nu)>-1)$. $\lim_{\nu\to+\infty}J_\nu(x)=0$ for any fixed $x$. I think the above two properties of ...
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116 views

Integral $\int_{0}^\infty\frac {(1-{{e}^{-i (q-p)t}})ln(|p^2-p_0^2|)}{(q-p)({{ p}}^{2}-{{p_1}}^{2})({{p}}^{2}-{{p_2} }^{2})}dp$

I am trying to get a closed form analytic result for the integral $$\int _{0}^{\infty }\!{\frac {\left(1-{{\rm e}^{-i \left( {q}-{p} \right) t}}\right){\rm ln}(|p^2-p_0^2|)}{ ( {q}-{p} ) \left( {{ ...
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136 views

How to perform this matrix integral?

Edit: some backgrouds added. In quiver matrix model which is reviewed DV or CKR, the path integral reduce to the matrix integral $$Z \sim \int \prod_{i=1}^r d\Phi_i \prod_{<a,b>} dQ_{ab} ...
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91 views

solve nonlinear second order ODE

I obtained Nonlinear second order differential equation as $y\cdot y''+y'^2-m\cdot y^{-a}y'^2+k=0$, Where $y'= \dfrac{dy}{dx}$, $y''=\dfrac{d^2y}{dx^2}$. I could not obtain the solution so please ...
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976 views

Integral of a random process that follows Gaussian Process

Suppose $X(t)$ follows a strictly-sense stationary(SSS) Gaussian Process with the mean to be $\mu$ and autovariance $\sigma^2$ How to prove that $\int_{0}^{T}{{X(t)}dt}$ is random variable that ...
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215 views

Integration of a function where the integral is equal to 0

Let $f:[a,b]\rightarrow \mathbb{R}$ be a continuous function. Suppose that $\displaystyle\int_a^b x^nf(x) \, dx=0$ for all $n\in\{ 0,1,2, \ldots \}$. Prove that $f=0$. (Hint. Consider ...
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68 views

infinite sum and integral representation

assuming all the numbers $ k_{n} $ are REAL is then true that $$ \sum_{n=0}^{\infty}\frac{1}{k_{n}^{s}}=\frac{1}{2 \Gamma (s) cos(\pi s)}\int_{0}^{\infty}dtt^{s-1}\sum_{n=0}^{\infty}cos(k_{n}t) $$ ...
3
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170 views

Integrating the exponential of a complex quadratic matrix

Problem statement I'm trying to do a discretized path integral/functional integral. The integral that I'm stuck with is of the form $$ \int_{-\infty}^{+\infty} \mathrm{d}^3\vec{x}_1\, ...
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144 views

How to integrate the following formula about normal distribution

How to compute the following formula? $$ \int_{-\infty}^{+\infty} \Phi(x) N(x\mid\mu,\sigma^2) \, dx $$ $$ \int_{-\infty}^{+\infty} \Phi(x) N(x\mid\mu,\sigma^2) \, xdx $$ where ...
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58 views

Find the surface integral of $f=|x|-|y|$ over the part of $z=1-\frac{x^2}{M}-\frac{y^2}{N}$ inside a cylinder.

(a) Find the surface integral of $f=|x|-|y|$ over the part of $z=1-\frac{x^2}{M}-\frac{y^2}{N}$ inside the region $\frac{x^2}{M^2}+\frac{y^2}{N^2}=1$ (b) Find the surface integral of $f=|xy|$ over ...
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100 views

$|f(t) - f(s) |\leq \int_s^t g $ then $f(t) - f(s) = \int_s^t h.$

Let $f : [0,1] \rightarrow [0, + \infty)$. If there exists $g \in L^1([0,1]) $ s.t. for every $t,s \in [0,1]$ holds $$ |f(t) - f(s)| \leq \int_s^t g(u) \, du \quad (t>s),$$ then there ...
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91 views

How to calculate hard integral?

How to calculate the integral $$ \int_D \frac {\prod_{i<j}(a_i-a_j)^2\prod_{i<j}(b_i-b_j)^2} {\prod_{i,j} (a_i+b_j)^2}\,d\lambda_{2n-1},$$ where $$D:=\{ (a_1,\dots ...
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156 views

Powers of the Meijer G function

I am trying to find the following integral: $$ I = \int_0^{+\infty} J_{1}(2\beta\sqrt s) \Psi(m,1-m,\beta^2/a)^L d\beta, $$ where $J_{1}(2\beta\sqrt s)$ is the known Bessel function, $\Psi$ is the ...
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188 views

Multivariable calculus: optimizing for shortest path along a curvy plane?

I want to write a computer program which can help me spend the least amount of energy and time walking between locations on my university campus. My campus is very hilly, and it is also extremely hot ...
3
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240 views

integral with dirac delta of a periodic argument

Is my solution of the following integral correct: $$I(\varphi,\varphi_0)=\int_{-\infty}^\infty x e^{-jx\cos(\varphi-\varphi_0)}\mathrm dx \qquad ?$$ I know that: $$\displaystyle ...
3
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191 views

What is $\int^{\frac{\pi}{2}}_0 (x -[\sin x])dx$ equal to ( where [.] denotes the greatest integer function)

Problem : $\int^{\frac{\pi}{2}}_0(x-[\sin x])~dx$ is equal to (where [.] denotes the greatest integer function) I have solved it the following way by separating two functions: i.e. $x$ and $[\sin ...
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70 views

Difficult Definite Integral with inverse Cosh

How can I solve this integral containing inverse cosh? Does it have any antiderivative? $$ \int_b^r t^2 \operatorname{arccosh}(a/t) \sqrt{r^2 - t^2} d t$$ for $0< b< r< a$.
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71 views

Vanishing of integral over hemispheres implies vanishing of function?

Consider a function $F$ on the half space $\{(x,y,z)|z>0\}$. If $F$ is analytic, it is straightforward to show that A) The integral of $F$ over the hemisphere $(x-x_0)^2 + (y-y_0)^2 + z^2 = R^2$ ...
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441 views

Integrating a fractional power of a rational function

I am currently working on a project where I stumbled upon the integral $$ \int \frac{\sinh \left(\frac{R}{2}\right)}{(\coth R - 6R \coth\left(\frac{R}{2}\right) + 9)^{1/4}} \,dR $$ where $R$ is a ...
3
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104 views

Double Integration.

I have an integral $$\int_0 ^a\int_0 ^b\int_0 ^a\int_0 ^b \sin(x)\sin(\bar{x})\sin(y)\sin(\bar{y})f(x,\bar{x},y,\bar{y})~dx~dy~d\bar{x}~d\bar{y}$$ where $f= ...
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179 views

Integral of a max function over a hypersphere; expected max z score

Is there a closed-form expression for the integral of max($x_1,..., x_n$) over the ($n-2$)-dimensional hypersphere, {$x \in \mathbf{R}^n$: $\sum_{i=1}^n x_i = 0$, $\sum_{i=1}^n x_i^2 = 1$}? I come ...
3
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108 views

Frullani version of the classic $\log \left( 1 + 2\alpha \cos px + \alpha^2\right)$ integral.

I read in a paper about Frullani integrals the following claim $$ \begin{align*} I & :=\int_0^\infty \frac{1}{x}\log\left(\frac{1 + 2\alpha \cos px + \alpha^2}{1 + 2\alpha \cos qx + ...
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305 views

Turning a Line Integral into a Contour one

I'm trying to compute an integral appearing in the article "On Determinants of Laplacians on Riemann Surfaces" of D'Hoker and Phong (page 541). It is as following. Fix $B\in \mathbb{R}_+$ and let ...
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85 views

Show that $\int_{\alpha}\frac{1}{z}\, dz=\int_{\beta}\frac{1}{z}\, dz$.

Let $a$ and $b$ be positive real numbers. Define ways $\alpha,\beta\colon [0,1]\to\mathbb{C}$ via $$ \alpha(t):=a\cos(2\pi t)+ia\sin(2\pi t),~~~~~\beta(t):=a\cos(2\pi t)+ib\sin(2\pi t). $$ Show ...
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92 views

What do I need to know to integrate any rational function?

My analysis book makes the following statement: Every rational function with real coefficients can be integrated in terms of rational functions, logarithm functions, arctangent ...
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222 views

Understand integral from Gradshteyn and Ryzhik book “Table of integrals, series, products”

I was checking useful integrals in this book. I have found one (6.298) that is what I need, but I don't understand how every step towards the final result works. ...
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510 views

spectral integral

I am learning spectral integration for my summer. I am stuck at a point. Having got hold of a spectral measure, we define the spectral integral of a simple function as usual and then approximate any ...
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32 views

Given $\Sigma$ a surface parameterized by $\Phi : D \to \Sigma$, prove a certain formula for $area(\Sigma).$

Let $\Sigma$ be a surface parameterized by $\Phi : D \to \Sigma$, and let $$A=\Phi_u \cdot \Phi_u~,~B=\Phi_u \cdot \Phi_v,~ C=\Phi_v \cdot \Phi_v.$$ Prove $$area(\Sigma)=\int\int_D \sqrt{AC-B^2} ...
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77 views

What is $\int_{-\infty}^{\infty} \frac{e^{-\alpha t} \cos[t + y]}{1+\beta e^{-2\alpha t} } dt$?

I want to compute the following integral: $\int_{-\infty}^{\infty} \frac{e^{-\alpha t} \cos[t + y]}{1+\beta e^{-2\alpha t} } dt$ with $\alpha, \beta, c$ real constants, and $\alpha>0,\beta=0$. ...
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118 views

A photon in expanding Universe (a snail on a tree)

I want to know how far a snail can reach in expanding universe. It has a constant speed c = 1 and tree is expanding at speed $v= H_0 D$, with Hubble constant $H_0 = 1$. Here D(T) is the distance of ...
3
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103 views

Differential equation $y'(t) = 1-y(t) e^{y(t)-1}$

I am interested in finding a clean explicit solution (if possible) to the differential equation $$ y'(t) = 1-y(t) e^{y(t)-1}, $$ where $0 \le t < 1$ and $0 \le y \le 1$. This can obviously be ...
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76 views

Hadamard finite part in 2 dimensions

given the divergent integral $$ \int_{0}^{\infty}dx \int_{0}^{\infty}dy \frac{xy+x^{3}-y^{2}}{1+xy+x+y} $$ how could i get a finite value in the sense of Hadamard finite part integral ?? thanks, ...
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147 views

computing a difficult integral using software

I'd like to compute the following integral. I've tried SAGE but it just runs for 15 minutes then stops.. not sure what that means. If anyone wants to take a crack with mathematica or anything, please ...
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124 views

Integral of the Normal Characteristic Function

The characteristic function of the $N$-variate Normal distribution is $$\forall \mathbf{t} \in \mathbb{R}^N \quad \psi(\mathbf{t}) \equiv \mathbb{E}\left( e^{i\mathbf{t}X}\right) = \exp \left( i{ ...
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278 views

Prove Heisenberg uncertainty principle (measure and integration theory)

Here is a question in measure and integration theory, Let $f$ be a continuously differentiable complex function on $\mathbf{R}$ s.t. the functions $x \mapsto xf(x)$ and $f'$ are in ...
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133 views

Integration by parts of a normalized function - [copied from Physics.SE]

By using integration by parts, I need to show for $$A = \frac{\mathrm d}{\mathrm dx} + \tanh x, \qquad A^{\dagger} = - \frac{\mathrm d}{\mathrm dx} + \tanh x,$$ that ...
3
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0answers
195 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 ...
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344 views

Under which hypotheses is switching between sum and integral signs legit?

Which hypotheses are needed to change the order of sum and integral signs? Concrete example: consider the expression $$ ...
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177 views

Simplifying an integral arising in Physical Chemistry

I am struggling to understand the following transition (encountered in a paper on Physical Chemistry). Let $$D=\frac{\tau_0^{-1}\int_0^\infty G(t)dt}{1-\tau_0^{-1}\int_0^\infty G(t)\int ...
3
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424 views

Is there a way to Fourier transform Cos[Sin[x]]

In my physics problem, I encountered a solution has the form like Cos[Sin[t]], and I need to do the Fourier transform to this solution. Is there a way to do the Fourier transform analytically to ...