All aspects of integration, including the definition of the integral and computing different types of integrals. For questions solely about the properties of integrals, don't use this tag alone! Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another ...

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10 views

A pseudometric on the space of the measurable functions is complete

I'm working in the following exercise: Suppose $(X, \mathcal A, \mu)$ is a finite measure space and suppose $\mathcal F$ is the set of all $\mathcal A$-measurable functions $f: X \rightarrow \mathbb ...
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0answers
3 views

How I integrate this function? with Delta Fuction

here is $$∫_0^l F(x,ζ)ϕ(x)dxdζ$$ with this $$F(x,ζ)=$$ $$Q(ζ)δ(x-x0) e^(-rx)$$ Thanks for your help!
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0answers
17 views

Closed form of integrals containing double exponentials

Are there closed forms for the following integrals? $$\begin{align} I_1(w) & = \int_{-\infty}^{\infty} \frac{\exp(-we^y)}{y^2+\pi^2} dy, \\ I_2(w) & = \int_{-\infty}^{\infty} ...
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0answers
27 views

How to take this Grassmann integral?

I'm trying to reconstruct and understand what is explained in a paragraph of this paper. I am trying to check if the method they describe actually gives us the Laughlin state. The integral I'm facing ...
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0answers
28 views

Evaluating convoluted integrals of complex exponentional and rational

I want to evaluate the following integral: \begin{equation} f_{abcd}(t) = \int_{-\infty}^{\infty}d\lambda\int_0^{t-\lambda} d\tau \frac{e^{i a \tau}}{ (b+i \tau)^{5/2} } \int_0^{t-\lambda} d\tau ...
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4answers
42 views

How to find the integral of $(1-|\tau|)\cos(\omega\tau)e^{-j\omega\tau}$

I have a function that need calculate the integral. Could you help me to find it. Thank you so much $$f(\omega)=\int_{-1}^1(1-|\tau|)\cos(\omega\tau)e^{-j\omega\tau}d\tau$$ where $\omega$ is constant. ...
2
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1answer
44 views

Switching $\int$ and $\sum$ proof

Been reading through this proof which seems incorrect: Let $f_n$ be continuous on the curve $C$ and $\sum f_n$ converge uniformly on $C$. Then $\sum\int_Cf_n(z)dz=\int_C\sum f_n(z)dz$ PROOF: ...
0
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2answers
73 views

How to write this integral in a nice way?

I have a function $f(a,b):= \int_{-1}^{1} e^{i (ax+bx^2)}dx$ with $(a,b) \in \mathbb{R}^2 \backslash \{(0,0)\}$ and now I want to find out what $|f(a,b)|^2$ is. Is there a way to write this in a ...
2
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1answer
36 views

$\int_0^{2 \pi} \cos(x)e^{i (a \cos(x) + b \cos^2(x)} dx$ and $\int_0^{2 \pi} \cos^2(x)e^{i (a \cos(x) + b \cos^2(x)} dx$

I am currently dealing with the two integrals in the title and I want to find out, when their real part of their imaginary part vanishes ( so for which constellation of $(a,b) \in \mathbb{R}^2 ...
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1answer
72 views

Compute polylog of order 3 at $\frac{1}{2}$

How to compute the following series: $$\sum_{n=1}^{\infty}\frac{1}{2^nn^3}$$ I am aware this equals polylog of order 3 at $\frac{1}{2}$, but how to prove it using integral or Euler sum only (without ...
2
votes
1answer
41 views

Integral of $\frac1{\cos^n x}$

Hi guys I have already proven for an assignment that: $$\int\cos(x)^n dx=\frac{1}{n}\cos(x)^{n−1}\sin(x) + \frac{n-1}{n}\int\cos(x)^{n−2}dx$$ Now we have been asked to calculate ...
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1answer
36 views

invariance of integrals for homotopy equivalent spaces

I just wanted to know whether the integral of a closed n-form is invariant if we integrate it over homotopy equivalent spaces. This seems like a generalization of "Homotopy invariance of line integral ...
2
votes
3answers
65 views

How to find the antiderivative of this function?

I want to integrate this function: $$\int\dfrac{x^2}{e^x-1}dx$$ I used integration by parts formula to integrate it. However I have reached somewhere where I got something like this: ...
2
votes
4answers
69 views

Integrating $x^3\sqrt{ x^2+4 }$

Trying to integrate $\int x^3 \sqrt{x^2+4 }dx$, I did the following $u = \sqrt{x^2+4 }$ , $du = \dfrac{x}{\sqrt{x^2+4}} dx$ $dv=x^3$ , $v=\frac{1}{4} x^4$ $\int udv=uv- \int vdu$ $= ...
2
votes
1answer
51 views

How find this integral $I=\int_{-\pi}^{\pi}\frac{x\cdot \sin{x}\cdot arccot{(2014^x)}}{1+(\cos{x})^4}dx$

Question: Find this integral $$I=\int_{-\pi}^{\pi}\dfrac{x\cdot \sin{x}\cdot arccot{(2014^x)}}{1+(\cos{x})^4}dx$$ let $x\to -x$,so $$I=\int_{-\pi}^{\pi}\dfrac{x\sin{x} \cdot ...
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1answer
17 views

Find the Laplace transform of integral(from 0 to x) sin(2t) dt

Find the Laplace transform of $\int_0^x\,\sin\,(2t)\,dt$ So basically, $$\int_0^x\,\sin\,(2t)\,dt = -\frac{1}{2}(\cos\,(2x) - 1)$$ So $$\mathcal{L}\{\cos\,(2x)\} = \dfrac{s}{s^2 + 4}$$ ...
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8answers
1k views

Really advanced techniques of integration (definite or indefinite)

Okay, so everyone knows the usual methods of solving integrals, namely u-substitution, integration by parts, partial fractions, trig substitutions, and reduction formulas. But what else is there? ...
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3answers
43 views

Calculus 2 Integral of$ \frac{1}{\sqrt{x+1} +\sqrt x}$

How would you find the integral of $1/(\sqrt{x+1} + \sqrt x) dx$. I used u substitution and got this far: $u = \sqrt{x+1}$ which means $(u^2)-1 = x$ $du = 1/(2\sqrt{x-1}) dx = 1/2u dx$ which means ...
1
vote
2answers
46 views

Integrate $\int\frac{dx}{x\sqrt{x^2+x+1}}$ [on hold]

Hello I need some help with the following integral: $$\int\frac{dx}{x\sqrt{x^2+x+1}}$$ Have been trying u-sub, and parts which do not get me to a solution!
2
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3answers
38 views

Integrating $\int_{-\pi}^{\pi} \frac{ d\theta}{w - sin \theta}$

I know that the integral $$\int_{-\pi}^{\pi} \frac{ d\theta}{w - sin \theta} = \frac{2\pi}{\sqrt{w^2-1}}$$ where w, is an arbitrary constant and at some point you must do the substitution $$u = tan( ...
1
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0answers
23 views

Simple complex line integral over a rectangle

What is the easiest way without using residues to calculate: $$\int_{\gamma} {\overline z \over {8 + z}} dz$$ Where $\gamma$ is the rectangle with vertices $\pm 3 \pm i$ in $\Bbb C$ in the clockwise ...
2
votes
2answers
31 views

Spectral Measure: Support

The support of a spectral measure is defined by: $$\mathrm{supp}E:=\bigcap_{C:E(C)=1}C$$ where $C$ are closed subsets (see german wiki). So by definition it is closed. However I'm wondering wether it ...
9
votes
1answer
99 views

Integral of $\sqrt{x^3 + 8}$?

I have issues solving the following integral: $$\int\sqrt{x^3+8}~dx$$ I tried substitution and integration by parts, but with no use. I'm guessing I have to use some trigonometric substitution. ...
0
votes
1answer
25 views

Calculating area relative to the y-axis

I was asked to calculate the area of the region bounded by the following graph: $$ y = x^2+4x ; y=0$$ I substituted $y$ in order to get $x = 0$ 0r $x=4$. Now I would like a little bit of help to get ...
1
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1answer
39 views

Spectral Measure Integration: Product

Given a Hilbert space $\mathcal{H}$ and spectral a measure $E:\Sigma(\Omega)\to\mathcal{B}(\mathcal{H})$. Define the integral of simple functions by: $$\int_\Omega ...
0
votes
1answer
50 views

Area enclosed by curves

Given the curves $y=x^2$ and $y=\frac{1}{2}(x+x^4)$. What is the area enclosed by them ? I can't find the points of intersection of the curves.
15
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1answer
123 views

Computing $\lim\limits_{n\to\infty} \Big(\sum\limits_{i = 1}^n \sum\limits_{j = 1}^n \frac1{i^2+j^2}-\frac{\pi}{2} \log(n)\Big)$.

In the chatroom we discussed about the asymptotic of $\displaystyle \sum_{i = 1}^n \sum_{j = 1}^n \frac1{i^2+j^2}$, and if we think of the inverse tangent integral, it's easy to see that ...
2
votes
1answer
26 views

Volume generated by lemniscate revolving about a tangent at the pole.

The lemniscates $r^2 = a^2\cos2\theta$ revolves about a tangent at the pole. What is the volume generated by it ? Please explain in detail. I found a couple of answers on finding surface areas, ...
6
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7answers
103 views

How to integrate $\int_{-\infty}^\infty e^{- \frac{1}{2} ax^2 } x^{2n}dx$

How can I approach this integral? ($0<a \in \mathbb{R}$ and $n \in \mathbb{N}$) $$\large\int_{-\infty}^\infty e^{- \frac{1}{2} ax^2 } x^{2n}\, dx$$ Integration by parts doesn't seem to make ...
2
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3answers
291 views

Evaluating the integral $ \int{\frac{x}{\sqrt{2x^2 + 3}}}dx $

I am trying to integrate the following: $$ \int{\frac{x}{\sqrt{2x^2 + 3}}}dx $$ It seems to me to be a trig substitution; however, I couldn't seem to get it into one of the three forms, i.e., ...
1
vote
1answer
56 views

Bochner: Lebesgue Obsolete?

Bochner's notion of integral: $$F\text{ Bochner integrable}:\iff \exists S_n\in\mathcal{S}:\quad \int\|S_m-S_n\|\mathrm{d}\mu\to 0\quad(S_n\to F)$$ This version totally circumvents Lebesgue's notion ...
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3answers
36 views

Triple Integral in Spherical Coordinates.

$\newcommand{\de}{\operatorname{d}}$A little stuck on this one. $$\iiint_V ye^{-(x^2+y^2+z^2)^2}\de V$$ Use Spherical Coordinates to evaluate where V is the solid that lies between y=0 and the ...
1
vote
2answers
45 views

Elias Stein : Real Analysis

I cannot understand why this particular line in the text is true: " Moreover, there are $O(k^{d-1})$ cubes in $\cal{Q}\ '$ " For the text see ...
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1answer
18 views

Integral on complex plane of a gaussian times power

I can't solve the integral $$ I = \int_\mathbb{R} \int_\mathbb{R} \ (x + i y)^{2k} \ e^{\displaystyle - \frac{(x + i y)^2 R^2}{1+R^2} - y ^2} d x d y $$ which can be rewritten as $$ I= \int_\mathbb{R} ...
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3answers
36 views

Solving using integrating factor [on hold]

Q) Solve $y' = 2x + y$ using the integrating factor. Can anyone guide me with steps here? Help appreciated. Thanks.
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0answers
35 views

Evaluating sums and integrals using Taylor's Theorem

Taylor's theorem states that $$f(x)-\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}x^k = \int_a^x \frac{f^{(n+1)} (t)}{n!} (x - t)^n \, dt $$ This could be used to evaluate partial sums using knowledge of the ...
3
votes
1answer
43 views

Calc 2: Integration by Parts w/ trig identities

$$\int e^{3\theta}\sec^4(e^{3\theta})\tan^{11}(e^{3\theta})d\theta$$ I just want to make sure that I'm doing this correctly so that I can understand the material. I would also appreciate any tips or ...
2
votes
3answers
83 views

Integral $\int_0^\pi \frac{x\,\operatorname dx}{a^2\cos^2x+b^2\sin^2x}$

Integrate: $$ \int_0^\pi \frac{x\,\operatorname dx}{a^2\cos^2x+b^2\sin^2x} $$
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0answers
42 views

Evaluating integral involving product of cosine inverse

I am trying to evaluate the below mentioned integral which involves product of two cosine inverses and two variables $x$ and $y$. I need to evaluate the integral or get an approximate value of this ...
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0answers
13 views

Generalized change of variables in integral

When I read the following (http://www.math.helsinki.fi/~analysis/GraduateSchool/maly/gs.pdf ), it is hard to understand it. In particular, what does it mean by the last equation? Why does it make ...
3
votes
5answers
234 views

Why consider square-integrable functions?

Why are $L^2$ functions important? From reading around I have three hypotheses: they show up in QM (but, why?) they form an inner product space (but, is that a "tight bound" or is the class easily ...
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0answers
60 views

Is there a formal proof of this basic integral property?

This has really been bothering me because everywhere I have looked the answer has been "A proof has been omitted because the theorem is very intuitive" or "Proofs are very complicated and not worth ...
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0answers
68 views

Prove there exist a $p$ so that the inequality holds

I am stuck with the following problem. Given the Gaussian mixture distribution $f(\cdot)$ $$ f(x) = ...
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0answers
50 views

Solution to the Integral

I am trying to solve a pdf which contains the following integral. The integral would involve the inverse of cosine function. Can anybody provide me the method how to solve the below mentioned ...
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3answers
49 views

Integration by parts: $\int e^{-\theta}\cos7\theta \;d\theta$

$$\int e^{-\theta}\cos7\theta \;d\theta$$ I started off by using $u=\cos 7\theta$ and$ \;dv=e^{-\theta}d\theta$, however, this just led me in a circle. I am now at: $$u=e^{-\theta},\;dv=\cos 7\theta ...
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2answers
35 views

Calculus 2: Strategy for Integration, Integral of e^(x+e^x)dx

How would you find $\int e^{x+e^x}dx$? I know I need to use $u$-substitution but I tried changing what I use for $u$ but I still could not get the right answer. If someone could push me in the ...
2
votes
1answer
45 views

A proof involving nested integrals and induction [duplicate]

Prove that $$\int_0^x dx_1 \int_0^{x_1}dx_2 \cdots \int_0^{x_{n-1}}f(x_n) \, dx_n =\frac{1}{(n-1)!}\int_0^x (x-t)^{n-1}f(t) \, dt$$ I'm trying induction over $n$. The case $n=1$ is trivial. When ...
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1answer
35 views

Integral Test for convergence of a series

"Consider the series given by $$\sum_{n=2}^{+\infty}\frac{1}{n\ln n(\ln(\ln n))^{\alpha}}$$ for $\alpha>1$. Use the Integral Testo to conclude if the series is convergent or not." I tried to make ...
1
vote
1answer
28 views

Lebesgue Dominated Convergence: Alternative Proof?

Is there an alternative proof of Lebesgue's dominated convergence theorem relying on positive functions only? The point is I'd like to prove that for positive functions: $$\int ...
0
votes
1answer
42 views

How to find $F(x) = \int_x^{x^2} (2+\sqrt t )\, dt$ ?

I have this problem: $$ F(x) = \int_x^{x^2} (2+\sqrt t )\, dt $$ I have to solve the integral. I got $2x^2+\frac{2x^3}{3}-2x-\frac{2x^{3/2}}{3}$ However, I don't think that it correct.