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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14 views

Asymptotics of integrals

I am struggling with this problem where we're asked to use method of steepest descent: Find the leading term of the asymptotics of the following integral for $\lambda\to\infty$ : ...
3
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2answers
19 views

Showing that $\lim \int \left(\sum_1^n |f_k|\right)^p \le \left(\sum_1^\infty \|f_k\|_p\right)^p$

I am reviewing a proof about the completeness of $L^p$ spaces. The proof begins as such (Folland Theorem 6.6): For $1 \le p < \infty$, suppose $\{f_k\} \subset L^p$ and $\sum_1^\infty \|f_k\| = ...
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1answer
16 views

What is this toeplitz like matrix called and how do I represent it as a convolution?

I have a matrix that is used to represent the Green's function in a popular class of fast E&M solvers (CG-FFT). The matrix represents distances, that are later filled in using the appropriate ...
1
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1answer
16 views

Mass and center of mass using double integrals

Disclaimer: This was given as a homework from college but the teacher didn't teach us anything about density or mass or anything related. A lamina has the form of the region limited by the parabola $ ...
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0answers
17 views

analysis, rieman, integration, real analysis

f is an integrable function in a compact set [a,b] -> R. Prove if integral((f(x)^2) over the set [a,b] is equal to zero then f(x0)=0 for any x0 in the compact. what you can say about the set {x in ...
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2answers
17 views

On the horizontal integration of the Lebesgue integral

I'm studying Lebesgue integral and its difference with respect to the Riemann one. I'm reading that the key difference (at least graphically speaking) is that the first slices the function ...
2
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2answers
33 views

Finding F ' (y)

This question looks deceivingly simple to me so I was wondering if someone out there could enlighten me whether it really is what I think it is or if I have completely missed the point. Let $$F(y) = ...
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0answers
17 views

Explain the formula of energy in signal processing [duplicate]

Please, give me intuitive understanding of this formula (http://en.wikipedia.org/wiki/Energy_%28signal_processing%29): So t is time, x(t) - signal function, integral is sum of this function on ...
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0answers
14 views

Heuristic: Daniell integral vs. Lebesgue integral

What are the advantages of the Daniel Integral over the Lebesgue integral and visa-versa? Heuristically speaking, I was wondering why this axiomatic operator is less popular besides the fact that it ...
1
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0answers
14 views

How does a Dirac delta function operate on a Fourier-Stieltjes integral?

Consider a stationary complex random function $\zeta(t)$ represented as a Fourier-Stieltjes integral $$\zeta(t) = \int_{-\infty}^{+\infty} e^{i\omega t}dA(\omega)$$ where $dA(\omega)$ is the random ...
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0answers
17 views

determining the fermi velocity via density of states

The problem is to determine the Fermi velocity for a fermion gas at absolute zero. the problem using integrating a function that looks like $$ v = \frac{4\pi V}{h^{3}} m^{3} \int_{0}^{\infty}{ ...
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0answers
20 views

Triple integral boundaries

If $W\subset \mathbb{R}^3$ is bounded by the planes $y+z=2, 2x=y, x=0, z=0$, what are the boundaries of $\int\int\int_W x dV$? How can I find the boundaries if I take $dV$ as $dydxdz$, $dxdzdy$ and ...
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0answers
22 views

Inverse Fourier transform of Gaussian

I need to calculate the Inverse Fourier Transform of this Gaussian function: $\frac{1}{\sqrt{2\pi}} exp(\frac{-k^2 \sigma^2}{2})$ where $\sigma > 0$, namely I have to calculate the following ...
2
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1answer
31 views

Evaluating an integral by dominated convergence theorem

I would like to know how to solve this two problems: a) $$ \lim_{n\to \infty}\int_0^n \left( 1-\frac{x}{n} \right)^{-n}\log{(2+\cos(x/n))} \, dx $$ b) $$ \lim_{n\to \infty}\int_0^{\infty} n e^{-nx} ...
2
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4answers
278 views

There is some strategy to solve an integral of this kind?

How to solve the integral $$\int\frac{\ln x}{\sqrt{1-x}}dx$$ and $$\int\sqrt{\frac{x}{x-1}}dx$$ I have no idea of how to deal with these integrals. It's the first integral I attempted. ...
1
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1answer
32 views

Integral involving CDF of a normal distribution

Can we evaluate the following integral ? $$\int_0^\infty x e^{-x^2} \Phi(ax+b)\,\mathrm dx$$ Here $\Phi(\cdot)$ is the cumulative probability distribution function of a standard normal random ...
2
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2answers
21 views

Finding the bounds of a solid for triple integrals

Ok, so I have an answer, most likely the wrong one. The question being asked is: Using polar coordinates find the volume of the solid bounded below by the $xy–plane$ and above by the surface $x^2 ...
1
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0answers
16 views

Integral of a funtion avoiding hypergeometric functions

I'm solving the following differential equation: $$uy''(u)+\gamma y'(u)+\frac{1}{u(1-u)}=0,~~\gamma=constant$$ For that, I transform this equation into a first orer one: $$uf'(u)+\gamma ...
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1answer
37 views

What do these questions mean?

Can someone please explain what these questions mean in simpler terms? 1)If g is continuous on $[a,b]$ and for all $ x ∈ [a,b] $ we have $ g(x) ≥ 0 $ and also $ g(x_o) > 0$ for some $x_o ∈ ...
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2answers
22 views

Separable ODE (with partial fraction integration)

Im stuck half way trying to solve the equation below. I tried using partial fraction integration and I think Im somewhere near. I need to express my answer as stated in the picture as well Thanks in ...
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1answer
23 views

Finding limits of integration for double integral

Given a region where the $x$ limits are $-1< x<1$ and $0< y<\sqrt{4-x^2}$, with the option of converting into polar coordinates, i.e. the function $(x,y)$ can be replaced by $r^2$. I'm ...
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1answer
16 views

Finding the volume of the following solid using triple integrals

Find the volume of the solid in the first octant bounded by the coordinate planes, the cylinder $x^2 +y^2 =4$ and the plane $z+y=3$. I found the integral bounds just fine. So I have $\int_{0}^{2} ...
2
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1answer
24 views

Evaluating area using an integral in polar coordinates

I am trying to find the area of a circle which is given by the polar parameterization $$r(\phi) = \cos\phi + \sin\phi.$$ I can evaluate it in 2 ways and don't know why I get different answers. First ...
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0answers
29 views

Finding an appropriate scalar

I have the following integral $$a_N\int_0^\infty{\frac{2e^x}{e^{2x}+T_{N-1}(2x)}}dx=1$$ Where $T_N=\sum_{k=0}^N\frac{x^k}{k!}$ I'd like to find the value for $a$ which makes this true for all ...
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0answers
29 views

How to convert infinite intergral to sum

How to convert Wiener filter formulas from integral to sum? They are for images therefore it must be possible to convert them to sums. Any help will be appreciated: I could not find much info on ...
0
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0answers
27 views

Solve this integral or at least find an upper bound?

Let $r,t>0$ be fixed. Let $a,b,c$ be numbers such that the following integral converges (I think $a,b,c>-1$ is OK). Then I would like to compute the following integral explicitly if possible or ...
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0answers
25 views

A Integral inequality.

For any positive integer $n \in {\mathbb{N}^ + }$, prove inequality $$\int_{ - \pi }^\pi {\left| {\cos \left( {\frac{{2n + 1}}{2}t} \right)} \right| \cdot \left| {\frac{1}{{\sin \left( {\frac{t}{2} + ...
2
votes
2answers
54 views

A problem in definite integral.

What will be the value of $a$ for which the integral $$\int \limits^{\infty }_{0}\frac{dx}{a^{2}+(x-\frac{1}{x})^{2}} =\frac{\pi}{5050}$$ where $a^{2}\geq0$ It seems like a standard integral but ...
0
votes
1answer
16 views

Integral and dominated convergence theorem

Let us define $g_n(x)= n\chi_{[0,n^{-3}]}(x)$. I am looking for help to answers the following problem $(a)$ Show that if $f$ $\epsilon$ $L^2([0,1])$ then $\int_0^1f(x)g_n(x)dx \rightarrow 0$ as $n ...
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2answers
30 views

Is the following Differential Equation undefined for given values of X & Y?

I have been presented with the following differential equation which I'm asked to solve, where $y=0$ when $x=\pi$. $$(y+1)\sin x\frac{dy}{dx} = (y^2+1)\tan^2x$$ I notice that $(y^2+1)$ may be ...
2
votes
1answer
28 views

find $\lambda$ such that the integral has a solution.

I have the integral equation: $u(x) = f(x) + \lambda \int_0^{\frac{1}{2}}u(y)dy$ I have to find $\lambda$ such that the integral has a solution. How to approach such problems?
2
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0answers
46 views

How to find this integral from Gaussian integral? [duplicate]

How to find $$\int_0^\infty e^{-ax^2-\frac{b}{x^2}}dx$$ using gaussian integral? I tried complete the square: $$-ax^2-\frac{b}{x^2}=-\left(\sqrt{a}x+\frac{\sqrt{b}}{x}\right)^2+2\sqrt{ab}$$, but what ...
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5answers
91 views

Where is the mistake in proving 1+2+3+4+… = -1/12?

https://www.youtube.com/watch?v=w-I6XTVZXww#t=30 As I watched the video on YouTube of proving sum of $$1+2+3+4+\cdots= \frac{-1}{12}$$ Even we know that the series does not converge. First I still ...
1
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2answers
32 views

Double integral over a parallelogram

I understand the general concept behind double integrals but do not understand how to change the coordinates linearly, and what to do from there. Find $$\int\int_P(x+y)dxdy$$ Where $$P$$is ...
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votes
1answer
28 views

Find the indefinite integral and check the result by differentiation

Find the indefinite integral and check the result by differentiation. I have worked all the problems just I am stuck and would like to check my answers. (1) $\int(x-x^2)dx$ (2) ...
0
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3answers
20 views

Integration with a variable in the terminals.

I know that in general, if we integrate over some defined values of $x$ and $y$ we find that, for a function $f(x,y)$ $$\iint f(x,y) \ dxdy=\iint f(x,y) \ dydx.$$ However, if we were to integrate, for ...
0
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0answers
21 views

Integrating Associated Legendre Polynomials

As part of a derivation for the question I asked here in Physics stackexchange, I am trying to calculate the following integral, but I am not sure how to proceed: ...
2
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0answers
47 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 ...
1
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2answers
43 views

What does the notation $\int_A$ mean, where $A$ is an event in a probability space?

I am used to seeing integral notation like this, which means the integral over the domain from a to b. $$ \int_{a}^{b} $$ But now I am looking at a statistics book that says "let A be an event" and ...
5
votes
2answers
211 views

Basic integration question.

I have the integral $$\iint x^2y^2 \ dx\,dy$$ but I am meant to evaluate it at the limits $0<y<1$ and $-2y<x<2y$. I am wondering what terminals of integration I should put in for $x$. Do I ...
0
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0answers
34 views

How to find the average velocity of blood?

The velocity $v$ of blood that flows in a blood vessel with radius $R$ and length $L$ at a distance $r$ from the central axis is $$v(r) = \frac{P}{4\eta L}(R^2 − r^2)$$ where $P$ is the pressure ...
0
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2answers
49 views

Exponential Integration [duplicate]

I don't know how to solve this equation: $$\int_0^\infty e^{-x} (x-a)^m dx$$ where $a$ is a constant and $m$ = $n+1$ Thanks in advance for your help.
3
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6answers
52 views

How can I prove that $ \int \text{sech}(x) ~ \mathrm{d}{x} = {\sin^{-1}}(\tanh(x)) + c $?

How can I prove that $$ \int \text{sech}(x) ~ \mathrm{d}{x} = {\sin^{-1}}(\tanh(x)) + c? $$ I don’t know how to prove this identity. Any help? I tried to multiply by $ \dfrac{\cosh(x)}{\cosh(x)} $, ...
1
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1answer
70 views

The convergence of $\sum_{n\geq 2}\frac{1}{n^{p}(\ln(n))^{q}}$ with two different tests.

Let $p,q\in\mathbb{R}$ and consider the series $\sum_{n\geq 2}\frac{1}{n^{p}(\ln(n))^{q}}$. i) Show by the comparison test, that the series is convergent if $p>1$ and divergent if ...
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2answers
72 views

What methods to use to integrate $\sqrt{1+t^4}$?

I have this integral to evaluate $$\int^x_1 \sqrt{1+ t^4}\, dt$$ I have tried substitution, trig identity and integration by parts, i don't have any answer. Can anyone explain the method I need to ...
-1
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0answers
16 views

Can someone explain how to find the surface area of the unit sphere using Fubini's Theorem? [on hold]

Can someone explain how to find the surface area of the unit sphere using Fubini's Theorem?
2
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0answers
17 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 ...
1
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1answer
20 views

Solving IVP by Laplace transform

I'm trying to solve an IVP with non-constant coefficients $$ y'' + 3ty' - 6y = 1, \quad y(0) = 0, \; y'(0) = 0 $$ Taking the Laplace yields $$ s^2Y + 3(Y + sY') - 6Y = \frac{1}{s}$$ $$ Y' + ...
0
votes
3answers
38 views

How do I calculate the limit of this integral from n to n+2?

I need to find the limit, as $n\to\infty$ of $\int_n^{n+2}e^{-x^3}dx$. I tried taking the integral using integration by parts but that doesn't work so now I'm stuck.
5
votes
3answers
79 views

How can I finish integrating $\int {\sqrt{x^2-49} \over x} $ using trig substitution?

$$\int {\sqrt{x^2-49} \over x}\,dx $$ $$ x = 7\sec\theta$$ $$ dx = 7\tan\theta \sec\theta \,d\theta$$ $$\int {\sqrt{7^2\sec^2\theta - 7^2} \over 7\sec\theta}\left(7\tan\theta \sec\theta ...