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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Derivative respect to integral

Did anybody knows how to take this derivative $\frac{d\int_{a}^{b}z\sqrt{x}df(z)}{dx}$? Is it correct answer $\frac{d\int_{a}^{b}z\sqrt{x}df(z)}{dx}=\int_{a}^{b}z\frac{1}{2\sqrt{x}}df(z)$? P.S. x ...
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Iterated integral and integrability

Hi everyone: Suppose $B_{1}$ and $B_{2}$ are balls in $\mathbb{R}^{m}$ and $\mathbb{R}^{m}$ (let's say for $m,n\geq2)$. Suppose that $f(x,y)$ is defined and measurable eveywhere. Beside, $$0\leq ...
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16 views

$\pi^2\int_{0}^{\infty}\frac{x\sin^4(x\pi)}{\cos(x\pi)+\cosh(x\pi)}dx=e^2\int_{0}^{\infty}\frac{x\sin^4(xe)}{\cos(xe)+\cosh(xe)}dx=\frac{176}{225}$

On my recent two posts on hard-looking integrals Marco Cantarini and Jack D'Aurizio proved them see Marco and Jack This is our final hard-looking integral that yield a rational answer. (1) ...
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1answer
23 views

Double integral of log squared

Reading up on integral equations, the text states as matter of fact that $$\int_0^1 \int_0^1\ln^2|x-y| \, dx \, dy < \infty$$ Wolfram|Alpha confirms that the double integral in fact evaluates to ...
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18 views

Showing $|I(\lambda)|\le C\lambda^{-N}$

Let $\lambda\in\mathbb R$ and $I(\lambda)=\int_{\mathbb R^n}e^{i\lambda\phi(\xi)}a(\xi)d\xi$, where $a\in C_c^{\infty}(\mathbb R^n)$ and $\phi\in C^{\infty}(\mathbb R^n)$, and assume $D\phi$ does ...
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2answers
44 views

$\displaystyle\sum_k f(k) \approx \int dk \ f(k)$

I am trying to understand the statement: $$\sum_{k\geq 0} f(k) \approx \int dk\ f(k).$$ When we do an integral, $dk$ is an infinitesimal and so we are roughly speaking, summing over all values of ...
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1answer
26 views

Integral of the function $\sqrt{|y^2-x|}$ on the domain $x^2\le y\le2$, $|x|\le1$

I'm trying to solve this: Find $\iint\sqrt{|y^2-x|}dxdy$ over $D$, where $D=${$(x,y)\in\mathbb R^2| x^2\le y\le2$ and $|x|\le1$} by using absolute value definition and checking the region D I ...
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1answer
31 views

How to integrate $\int \arctan(e^{-\pi y/b}) \, dy$?

How to integrate $\int \arctan(e^{-\pi y/b}) \, dy$
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2answers
75 views

Integrate $\int e^x\frac{1+\sin x}{1+\cos x}dx$

Integrate $$\int e^x\cdot\frac{1+\sin x}{1+\cos x}\,dx$$ My try; First step: I let $$\frac{1+\sin x}{1+\cos x} = u$$ $$e^x = v$$ and then I applied integration by parts: $$\frac{1+\sin x}{1+\cos ...
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1answer
59 views

what does $\frac{\text{d}x}{x}$ mean?

I saw in a lecture recently the Gamma-function written like $$\Gamma (k) = \int_0^\infty e^{-x} x^k \frac{\text{d}x}{x}$$ and the professor said, that the integral was with respect to the measure ...
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3answers
65 views

Integrate $\displaystyle\int \dfrac{e^x}{1+\cos x}dx$

Integrate $\displaystyle\int \dfrac{e^x}{1+\cos x}dx$ My Effort; I couldn't nothing .
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7 views

Proof: Majorant- and minorantcriterion for convergence of improper integrals

Let $I$ a interval and let $f,g:I\to \mathbb{R}$ continuous with $0\le g(x) \le f(x) \forall x \in I$. Prove this propostitions: (a) If $\int_{0}^{\infty} f(x)dx$ convergent $\Rightarrow$ ...
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1answer
42 views

A closed form of a integral with exp and cos

Can we find a closed form for the following integral: $$\int_0^{\infty} \frac{e^{-x} \cos x}{1+x} \, {\rm d}x$$ No matter how hard I tried I cannot tackle it. I am pretty much afraid that if a ...
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2answers
32 views

If the left Riemann sum of a function converges, is the function integrable?

If the left Riemann sum of a function over uniform partition converges, is the function integrable? To put the question more precisely, let me borrow a few definitions first. Pardon my use of ...
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1answer
60 views

Hints for $\int\frac{1}{1 + (\log x)^2}\, d(\log x)\,dx$.

I can't start with this question please help, how to deal with $d(\log x)$ and $dx$. $(19)$ $$\int\frac{1}{1 + (\log x)^2}\, d(\log x)\,dx = \_\_\_\_\_\_\_\_+c.$$ (a) $\frac{\tan^{-1}(\log ...
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2answers
80 views

Does $\int_{0}^1\frac{\ln(1+x+x^2)}{x}\mathrm dx$ have a closed form?

$$\int_0^1 \frac{\ln(1+x+x^2)}{x} \mathrm{d}x = 1.09662$$ I am trying to find a closed form of this integral. I think one might exist, this integral looks like it might be related to $\pi$, but ...
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2answers
31 views

For every integer $n\ge 0$ $\int_{0}^{1}\frac{x^n-(-1)^n(x-1)^n}{x^n+(-1)^n(x-1)^n}dx=0$ Why?

Can someone explain to me, why is this integral always has a zero answer? n is an integer, $n\ge0$ $$\int_{0}^{1}\frac{x^n-(-1)^n(x-1)^n}{x^n+(-1)^n(x-1)^n}dx=0$$
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1answer
21 views

Complex Solids of Revolution

I know that to compute a solid of revolution of a function $f(x)$ rotated around the $y$-axis, one method we can use is the "shell" method. For example, $f(x)=1/4x^2\in [2,4]$, rotated around the ...
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1answer
80 views

Alternate proof of the integral: $\int_0^1 x^x(1-x)^{2x}\,dx\neq3/8$

I am looking into the integral: $$I=\int_0^1 x^x(1-x)^{2x}\,dx\neq\frac{3}{8}$$ How might you prove this to be true? What's tough is that the integral $$3/8\lt I<0.37503$$ numerically. I managed ...
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36 views

Let $f_n$ uniformly convergent to$ f$. Proof that $f$ is integrable and $\int_A f_n \rightarrow \int_A f$

(When I write integrable I mean Riemann-integrable) Let $A \subseteq \mathbb{R}^m$ be a block, $f_n:A\rightarrow R$ be integrable functions and $f_n \rightarrow_{un} f$. Proof that $f$ is integrable ...
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39 views

Evaluating a limit of an integral

I have a function $f(x,y,z) :\mathbb{R}^3 \rightarrow \mathbb{C}$, a smooth function. I know that $$ I = \int_{z \in \mathbb{R}}\int_{y \in \mathbb{R}}\int_{x \in \mathbb{R}} f(x,y,z) \ dx dydz $$ ...
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2answers
32 views

Finding the coefficients of a triangular wave.

I have the following equation that I want to solve $$a_k = \color{blue}{\frac{1}{T} \int_{0}^{T/2} 2 \frac{t}{T} e^{-i \frac{2\pi}{T}kt} dt} + \color{red}{\frac{1}{T} \int_{T/2}^{T} 2 \frac{T-t}{T} ...
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4answers
53 views

Solution of a differential equation involving $3$ variables $x,y,t$

QUESTION: Solve the differential equation- $$\frac{dx}{dt}+\frac{dy}{dt}+2x+y=0$$ $$\frac{dy}{dt}+5x+3y=0$$ I am unable to progress in solving these equations. For by any manipulation, I am ...
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4answers
45 views

Are these integrations the same?

For the integration of: $$v^2 + 6k^2 = -Mv \frac{dv}{dx}$$ I rearranged to get: $$\int \frac{1}{-M} dx = \frac{1}{2} \int \frac{2v}{v^2 + 6k^2} dv$$ Is this the same as the following integral that is ...
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1answer
33 views

Evaluating $\int R(X)sin(x) dx$ with residue theorem.

The integral I am trying to evaluate is: $$I = \int_{-\infty}^\infty \frac{x}{1+x^2}\sin x\ dx = \int_{-\infty}^\infty f(x)\ dx$$ The standard approach to this is to realise $\sin x$ as the complex ...
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1answer
47 views

Finding this line integral, on a sphere radius $a$

$$ c \equiv \left\lbrace\left(x,y,z\right)\quad |\quad x^{2} + y^{2} + z^{2} = a^{2}\,,\quad x + y + z = 0\right\rbrace $$ $$ \mbox{Find}\quad \int_{c}x^2 $$ This is a line integral. ...
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0answers
23 views

Spherical Harmonic integration

How do I solve the lower integration of spherical harmonics, in terms of Clebsch Gordon coefficients? \begin{equation} \int d\Omega Y^{l_1}_{m_1}(\hat n)Y^{l_2}_{m_2}(\hat n){Y^{l}_m}^*(\hat n) ...
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3answers
75 views

Evaluating an integral using the gamma function

My question regards an integral $$\int_0^\infty \frac{\sin(x^p)}{x^p}\mathrm{d}x$$ The answer should be $$\frac{1}{p-1}\cos(\frac{\pi}{2p})\Gamma(\frac{1}{p})$$ and I roughly know that I should apply ...
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0answers
32 views

Trigonometric integral $\int_{[-\pi,\pi]^2}{\frac{1-e^{-in\cdot\theta_1}}{1-\cos(\theta_1)\cos(\theta_2)}\,d\theta_1\,d\theta_2}$

I am trying to compute the following integral (see here). Since it seems to be the wrong approach, I am trying to calculate another one which I hope it will give me what I am looking for. My point is ...
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32 views

How to integrate $\int \left|f(x) +g(x) \right|^2dx$? [on hold]

I am dealing with a quantum mechanics exercise at which I need to find the probabilty of $\left| \psi \right |^2$. $\psi$ is composed of 2 real value functions, say $f$ and $g$. So generally, how to ...
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3answers
32 views

Showing the integral $\int_{\mathbb{R}} \int_{\mathbb{R}} \min\{ 1, (\max \{ |x|,|y| \})^{-3} \} dx dy$ converges

I am trying to bound the following integral: $\int_{\mathbb{R}} \int_{\mathbb{R}} \min\{ 1, (\max \{ |x|,|y| \})^{-3} \} dx dy$. I am very sure this integral converges, but whatever I try seem to ...
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1answer
34 views

If $f:[a,b]\rightarrow R$ is a uniformly continuous function then its absolutely continuous?

If $f:[a,b]\rightarrow R$ is a uniformly continuous function then is it true that $f$ is always absolutely continuous?
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15 views

Fix a typo involving the Lobachevsky function in Thurston's notes

I believe that there is a typo in these great notes Thurston's Three-Dimensional Geometry and Topology, Volume 1 (Princeton University Press, 1997), Chapter 7 that is provide us by MSRI, in the ...
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0answers
36 views

Sum of all sine harmonics

I was discussing this with my calculus teacher, but she didn't come up with anything. I would like to take an infinite sum of functions (sine specifically) but don't know how to do that. I would ...
3
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1answer
29 views

What is an example of a uniformly continuous function but not absolutely continuous

Is there a function that is uniformly continuous function but not absolutely continuous. My answer is $f(x)=x^{2}, \forall x\in R$ Is this right? Are there any other?
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2answers
72 views

Solve integral $\int \frac{x+1}{x^2-2x+5} dx$

I need to solve: $$\int \frac{x+1}{x^2-2x+5} dx$$ I cann see that $D>N$ so I tried to scompose the $D$ but I get: $$x_{1,2} = \frac{2 \pm \sqrt{4-20}}{2}$$ So $\Delta < 0$ and I tried to use ...
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1answer
27 views

Find $\int_{C}{\bf{F}}\cdot d{\bf{s}}$ through the line segment

Let $F=\left[\frac{x}{x^2+y^2},\frac{y}{x^2+y^2}\right].$ Let $C$ by the curve consisting of the line segments from $$(-1,0)\to (0,-2)\to (2,0)\to (3,4)\to (0,5)\to (-1,0)$$ Find ...
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11 views

$\int(\int\phi(a-z)dz)dz=\Phi(a-z)$

Lets assume $\phi(a-z)$ is integrable. Can I conclude that the following integral $$\int\left(\int\phi(a-z)dz\right)dz$$ Can be expressed by a function $$\Phi(a-z).$$ So in result: ...
3
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3answers
43 views

Why is $\frac{1}{x}$ not Lebesgue integrable on $[0,1]$?

My teacher said (without explaining) that $\frac{1}{x}$ is not Lebesgue integrable on $[0,1]$? Could someone please explain why is this true?
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1answer
25 views

Triple integral $\int_{0}^{2\pi} \int_{0}^{2\cos(\theta)} \int_{0}^{\sqrt{2r\cos(\theta)}} r \ dzdrd\theta$ to find volume of a solid

On evaluating the volume between $$x^2+y^2 = 2x\\z^2=2x$$ I set up the triple integral $$\int_{0}^{2\pi} \int_{0}^{2\cos(\theta)} \int_{0}^{\sqrt{2r\cos(\theta)}} r \ dzdrd\theta$$ for which ...
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1answer
51 views

Sophomores dream

On wiki there is a proof of Sophomore's dream. I am trying to understand what they did when changing the variable and how they got $e^{-u}$.
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2answers
60 views

Proving that two branch cuts can cancel out

Define the following functions $\mathbb{C}\to\mathbb{C}:$ $$u(z)=\frac{\log \left(z+\frac{1}{2}\right)}{z}\quad \left[-\pi\leqslant\arg \left(z+\tfrac12\right)<\pi\right];\quad v(z)=\frac{\log ...
3
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2answers
97 views

Possible to do better than an upper bound for$\int^{\infty}_0 e^{-x}\log(x)\ dx$?

I used the series expansion of $e^{-x}$ and the fact that $\log(x)$ was less than $x$ in the $(0, \infty)$ to get an upper bound and so use simple comparison to show this was indeed integrable over ...
3
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1answer
70 views

Show that $\int h_n'(x) \varphi(x)\, dx \to \langle \delta, \varphi\rangle$ - Generalized functions theory

In the book Partial Differential Equations by Robert Strichartz, there's an exercise (#$1$, page $9$) that I am not quite sure how to solve. Is there anyone could give me the principal steps how to ...
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5answers
152 views

Evaluating the definite integral $\int_0^3 \sqrt{9- x^2} \, dx$

I have been having a problem with the following definite integral: $$\int_0^3 \sqrt{9- x^2} \, dx $$ I am only familiar with u-substitution and am positive that it can be done with only that. Any ...
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2answers
65 views

show that $\int_{0}^{1}\frac{1}{x^2}\ln\left[\frac{(1+x^2)^2}{1-x^2}\right]dx=\pi$

Most integrals involved $\ln(x)$ seem to produced results of $\pi^2$, $\sqrt\pi$, $\pi\ln(2)$ etc, but rarely $\pi$ on its own. Here is one (1) ...
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1answer
44 views

Prove $\int_{0}^{1}\frac{1}{x^2}\ln\left[\frac{(1+x^2)^2}{1-x^2}\right]dx=\pi$

Most integrals involved $\ln(x)$ seem to produced results of $\pi^2$, $\sqrt\pi$, $\pi\ln(2)$ etc, but rarely $\pi$ on its own. Here is one (1) ...
8
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4answers
166 views

How does one integrate $x^2 \frac{e^x}{(e^x+1)^2}$?

How can I show this? $$ \int_{-\infty}^{\infty} x^2 \frac{e^x}{(e^x+1)^2} dx = \pi^2/3$$ I tried applying residuals, but the pole is of infinite(?) order.
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1answer
21 views

Doubts in Volume, Hypervolume in $R^4$

Recently I was reading about triple integrals and I came across the statement - "We saw that a double integral could be thought of as the volume under a two-dimensional surface. It turns out ...
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1answer
31 views

How to solve simultaneous inequalities (reasked)? [duplicate]

I am doing multivariable calculus, and specifically double integrals. I am facing difficulties finding the domain of the integal, however i am given the following equations: $$1≤2x+y≤2$$ $$0≤x−2y≤1$$ ...