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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2
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64 views

Gamma function still hard for me

During my study I find a form for gamma function it was $\Gamma (x) = \lim_{n\to\infty} \frac{n! n^{x-1}}{x(x-1)(x-2)........(x+n-1)}$ And then by simplify this limit I get $$\lim_{n\to\infty} ...
2
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34 views

Smart coordinates for six-dimensional integral

I have a (hopefully) simple question: I am dealing with a definite (on all of $\mathbb{R}^6$) six-dimensional integral $$\int_{\mathbb{R}^6} F(\vec{x}_1,\vec{x}_2)d^3x_1d^3x_2$$ where the function ...
2
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0answers
48 views

For which polynomials $P$ the integral $\int_0^\infty x^{z-1} P(x)^{-s} dx$ is computable?

I consider the following integral: $$ I(z,s)=\int_0^\infty \frac{x^{z-1}}{(P(x))^s}dx, $$ where $P(x) = a_0 + a_1 x + \cdots + a_n x^n$ is a polynomial of degree $n \geq 2$ with $P(x) > 0$ for ...
2
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0answers
32 views

Is this type of definite integral defined?

I was trying to generalise the definite integral in the following way: Let $f:[a,b] \to \mathbb{R}$ be a continuous function, and let $\phi:(-\epsilon,\epsilon) \to \mathbb{R}$ be defined in some ...
2
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0answers
13 views

Integral over $B_1^n(0)$

Evaluate $I = \int_{B_1^n(0)} (a_1x_1 + \cdots + a_nx_n)^{2/3}$ where $a_j \in \mathbb{R}$. Here's where I'm at, following a hint: Consider an orthonormal transformation $T$ with the first row equal ...
2
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0answers
27 views

Real version of Harish-Chandra-Itzykson-Zuber integral

I'm interested in an integral of the form $$ \int_{O(d)} \exp\left(-\frac{1}{2}\mathrm{trace}(CUAU^T)\right)dU $$ where the integration is with respect to the Haar measure on the orthogonal group, ...
2
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0answers
41 views

Hadamard finite part of an integral

How does one take the 'Hadamard finite part' of an integral? I am following a paper and the result stated is that $$\int_{0}^{\infty}U_{B}^{-2}(Y)-U_{B}^{2}(Y)\,\textrm{d}Y=-2.7950.$$ The function ...
2
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64 views

Numerical integration with matrices

I have a matrix integration problem. It is based on the first integral under the section, "energy transfer efficiency and transport time" in the article, environment-assisted transport. There is a ...
2
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0answers
56 views

Evaluate of $\int \frac{(a+bx)^k}{x} dx$

Consider the integral $$ \int \frac{(a+bx)^k}{x} dx. $$ Does this have a representation using (preferably) elementary or special functions? Edit: $0 <k<1$. Wolfram alpha produces an ...
2
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49 views

Differential equations, theoretical question.

$$ L(x)=x^{(n)}+a_1(t)x^{(n-1)}+\cdots +a_{n-1}(t)x'+a_n(t)x=b(t);\qquad a_1(t),a_2(t),\ldots\in C$$ $$U_j(\varphi)= \sum_{k=0}^{n-1}(M_{jk} \varphi^{k}(\alpha)-N_{jk} \varphi^{k}(\beta))= ...
2
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38 views

Separating the Complex Error Function into Real and Imaginary parts

I'm trying to do a numerical integral of the following form: $$\int_a^b (\mathbb{R}\left[\operatorname{erfi}(z)\right])^2 \, dz$$ That is, I would like to integrate the square of the real portion of ...
2
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0answers
28 views

Fubini´s Application

I don't think this should be difficult, but I can't prove this equality. I need to show that $$ \int\limits_0^a\int\limits_0^x\int\limits_0^y f(z) \;\mathrm{d}z\;\mathrm{d}y\;\mathrm{d}x = ...
2
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0answers
36 views

There is no quasiconformal map from punctured ball into ring (on the plane)

The idea is to use l2 cohomology as a quasiregular map invariant. It is easy to see that there are closed 1-forms on the ring which are not exact, but it occures that every closed l2-form in punctured ...
2
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0answers
30 views

Parametrization of surfaces for vector integration

I'm having some trouble calculating vector fields through surfaces. After attempting a few and being dissapointed with a wrong answer multiple times I figured I must be doing something wrong in the ...
2
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0answers
40 views

Finding area of a spheroid

Let $M=\{(x,y,z)\in \Bbb{R}^3 : (x/a)^2 + (y/b)^2 + (z/c)^2 = 1\}$. Find $\text{vol}_2(M) = \int_M 1 dS$. My attempt: The map $$\Phi:(0,\pi)\times (0,2\pi)\to \Bbb{R}^3\\ \qquad (\varphi, ...
2
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98 views

upper bound of a differential equation solution

Let $A(t)$ be a bounded singular values matrix that is function of time, and $f(t)$ an $L^\infty$ function of time. And consider the ODE $$ \dot x = A(t) x + f(t) $$ How we can describe qualitatively ...
2
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0answers
48 views

Laplace transform of inverse error function

I want to calculate the convolution of a function with the inverse error function. Therefore I chose to try to first find an integral transform of the inverse error function like the laplace ...
2
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0answers
11 views

When can we move a Fréchet derivative under a Lebesgue integral?

Under what conditions can we move a Fréchet derivative under a Lebesgue integral? Specifically, when does $$ G'(x) = h\in X\mapsto \int_{\Omega} \left(F_x^\prime(x,t)h\right) \mu(dt) $$ where $$ ...
2
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0answers
101 views

Geometric proof for Sophomore's dream

Is there a "visual proof" for sophomore's dream? $$\int_0^1 x^{-x}\,dx = \sum_{n=1}^\infty n^{-n}.$$ In the wikipedia article there are two algebraic proofs, but the integral and the summation has ...
2
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0answers
32 views

double integration with the same variable

I have the integral that I want to resolve. To calculate the flux of the electric machine, I have the following formula: $v_s= R_s \cdot i_s + \frac{\Phi _s}{dt}$ where $v_s, i_s, \Phi _s$ are ...
2
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26 views

Indefinite Hypergeometric Integral Transformations

I'm attempting to solve the indefinite integral $$S\left(v\right) = 2a\sqrt{\alpha ...
2
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47 views

Gamma Function Representation

I have some questions relating to the gamma function. My question is simply to evaluate the integral $$\int_0^\infty {t^{z-1}\sin t}\,dt$$ whenever $-1<\text{Re}(z)<1$. If we take $z$ ...
2
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41 views

Integration involving exponential function

I am trying to find the following integral $\int_t^s e^{-\frac{(\sqrt{x^2-a^2}-u)^2}{\sigma^2} } e^{-\frac{b^2}{x^2}} dx$ where $t<u<s$, I can find a solution for $u=0$, but I need the ...
2
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0answers
71 views

Strict inequality in proof of First Mean Value Theorem for Integrals

Let $M=\sup|f(x)|$ and $m=\inf|f(x)|$ $x \in [a,b]$ The first mean value theoremn for Riemann integrals says: If $f$ continuous (and in this case we will asume non-constant, constant is trivial) ...
2
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0answers
52 views

How to integrate $\frac{1}{\sqrt{x^2+y^2+z^2}}$

want to evaluate $$\int\frac{1}{\sqrt{x^2+y^2+z^2}}dxdydz$$ over entire $\mathbb{R}^3$ except $(0,0,0)$. I did this using polar coordinate and got ...
2
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0answers
23 views

What is an intuitive/geometric definition of line integrals? Do they work in 2-dimensions?

I understand that we are finding the area of a curve given by some function f(x) over the area of another curve C. (I've also successfully plugged and chugged my way through my homework, without ...
2
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0answers
51 views

Differentiation Theorem

Assume that a function $f$ is integrable on $[a,b]$ w.r.t. an increasing function $g$, that $f$ is continuous at $c\in[a,b]$ and that $g$ is differentiable at $c$. Then the function defined by ...
2
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133 views

Confused about trying to find the correct spherical co-ordinates for this tricky triple integral

I'm having trouble trying to figure out how to change the limits of integration to spherical co-ordinates in this particular question. I was wondering if someone would kindly be able to assist me in ...
2
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90 views

Finding $\int \frac{\sin\sqrt{\frac{x}{2}}}{\sqrt{x\cos\sqrt{x}}}dx$

Finding $$\int \frac{\sin\sqrt{\frac{x}{2}}}{\sqrt{x\cos\sqrt{x}}}dx$$ This is a homework. I tried to solve it by assuming $x=u^2$ but after that the integrals become not simple, so I don't know how ...
2
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0answers
50 views

On a problem about Rolle's theorem

Let $f:[1,3]\to\mathbb R$ be a continuous function such that $\int_1^2 f(x)dx=2$, and $\int_1^3 f(x)dx=3$, then there exists a real number $c\in(2,3)$ such that $$ \int_1^c f(x)dx=cf(c) $$ Note. I ...
2
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0answers
58 views

Double Integral of an Exponential Function with an Absolute Value in the Numerator of the Exponent

This is a question related to statistics, but my major concern relates to the setup and evaluation of integrals. So I decided this question was better suited for Mathematics Exchange than CV. I know ...
2
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0answers
37 views

If $A$ and $f$ are bounded, then $f$ is integrable in the extended sense (?) [Spivak]

I have a problem with one of the theorems in Spivak's Calculus on Manifolds. I will give some background first: An open cover $\mathcal{O}$ of an open set $A \subset \mathbb{R}^n$ is admissible if ...
2
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0answers
20 views

An extension of change of variables in double (and $n$-?) integrals - second-order Jacobian?

I'm aware that there are many, many questions regarding changing variables in double and triple integrals. The equation that typically pops up in textbooks is \begin{align} ...
2
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34 views

Proving a result using Riemann-Stieltjes integration

Let $(\alpha_n)_{n=1}^\infty$ be a sequence of monotonically increasing functions on $[a,b]$ such that the series $\sum_{n=1}^\infty \alpha_n(a)$ and $\sum_{n=1}^\infty \alpha_n(b)$ converge. I must ...
2
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0answers
68 views

Is this integral is right or wrong?

We did this exercise in class in a way, but at home I tried to solve it in a different way and I do not know if it is right or wrong. May you help me please? $\mathbf{\int tan^{5}x \, \, \, sec^{4}x ...
2
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0answers
175 views

Improvement of an Inequality

It would nice if someone could help me with this problem. I am looking at an improvement to the classical Jensen's Inequality: $$\int_\limits{}^{} \phi(x) \mu \mathrm{d}x \geq ...
2
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0answers
49 views

Prove $\iint_R f_{xy}(x,y)=f(a,c)-f(a,d)+f(b,d)-f(b,c)$

I'm asked to show that if $f_{xy}$ is Riemann Integrable on $R$, then $$\iint_R f_{xy}(x,y)=f(a,c)-f(a,d)+f(b,d)-f(b,c)$$ Where $R=[a,b]\times[c,d]$. To do this I can use the following formulation ...
2
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0answers
37 views

Separable equation

I am looking at this first order separable differential equation, and I am stuck. Here is the equation: $\frac{du}{dt}=u$ I seperated like this: $ \frac{du}{u}=dt $ Integrated both sides and ...
2
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0answers
39 views

Triple Integral - Varying Order of Integration

I need to express the triple integral for the volume of the solid enclosed by the surfaces $x = 4 - y^2$ and $y + z = 2$ in the first octant in $6$ different ways. I managed to get the correct answer ...
2
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0answers
19 views

Evaluate the following integral involving sign function

I = $\int_0^{x_1}\int_0^{x_1}\text{sgn}(y-x) x^{\alpha+i-1}e^{-x/2}y^{\alpha+j-1}e^{-y/2}dx dy$ where, $\text{sgn}(x) = 1$ if $x>0$ and $\text{sgn}(x) = -1$ if $x<0$. Otherwise it is zero.
2
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0answers
30 views

Integration over bounded regions

I have to evaluate the following integral \begin{align*} \int_Ee^{x^2-2xy+10y^2}dA, \end{align*} where $E=\{(x,y)\ |\ x^2-2xy+10y^2\leq1\}$. I already evaluated the integral and got ...
2
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0answers
44 views

Volume of paraboloid

Consider a paraboloid $x^2+y^2+2=z$. The task is to find the volum of the body obtained by confining the paraboloid by several planes $x=0, z=0, y=0$ and $x=3, y=3$. The zero planes cuts out a ...
2
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0answers
41 views

compute the complex-valued integral for the branch cut

Let $C$ be the circle of radius $2$ centered at origin. Let $f(z)$ be the branch cut of the function $z^{2−i}$ on the domain $−π < θ < π$. Compute the integral $$ \int_C f(z) dz$$ My attempt: ...
2
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0answers
43 views

Line integral - new parametric equation

We know that $$\int_\gamma V \cdot dr = \int_a^b V(r(t)) \cdot r'(t) dt$$ with $V$being our vector field and $r$being the parametric equation for the curve $\gamma$. Let now $\hat{r} = r \circ \phi$ ...
2
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94 views

$ \int \frac{\sin^{a}x}{\sin^{a}x+\cos^{a}x}dx$ and $ \int \frac{\cos^{a}x}{\sin^{a}x+\cos^{a}x}dx$

I tried solving this integral: $$\int^{\pi/2}_{-\pi/2} \frac{1}{2007^x + 1}\frac{\sin^{2008}x}{\sin^{2008}x+\cos^{2008}x}dx$$ I took a while before aptly applying the following identity I had noted ...
2
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0answers
61 views

Difference between Riemann-Stieltjes and Darboux-Stieltjes integral

Usually, analysis textbooks do not classify Riemann-Stieltjes and Darboux-Stieltjes integral. But I know that a function is $\alpha$-DS integrable does not implies it is $\alpha$-RS integrable. Can ...
2
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56 views

relationship between $\sin$ and $\cos$ integrals

I am trying to solve a physics problem and I end up with the following two equations. $\int\limits_{0}^{T}\sin\theta(t)dt = a$ and $\int\limits_{0}^{T}\cos\theta(t)dt = b$ The exact dependence of ...
2
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0answers
54 views

A tough double integral

I am looking at an ugly ratio of two random variables and am interested in the density of the the ratio. So what I did was to write down the joint characteristic function of the numerator and ...
2
votes
0answers
50 views

Integration imaginary and real part with branch cut

I have some problems with this integral $$ I=\int_{0}^{1}z(1-z)log(1-z(1-z)\frac{q^2}{m^2})dz $$ I see $z(1-z)$ get max value at $\frac{1}{4}$ and if $q^2>4m^2$ log function will be negative and ...
2
votes
0answers
47 views

closed form for integrals involving error function

Let $F(x)=\int_{-\infty}^xf(t)dt$ where $f(t)=\frac{1}{\sqrt{2\pi}}\exp{(-\frac{t^2}{2})}$. Then is there any way to calculate $$\int_{-\infty}^{\infty}F(x)^kf(x)^2dx$$for $k=2,3,...$ I started with ...