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Questions tagged [integration]

For questions about the properties of integrals. Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that describe the type of integral being considered. This tag often goes along with the (calculus) tag.

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is there a function $\phi$ such that $\int_R \int_R f(t)g(s)\phi(t,s)dsdt=\int_Rf(t)g(t)\rho(t)dt $?

Let $\rho$ be a positive function, is there exist a function $\phi$ such that for any $f,g$: $$\int_R \int_R f(t)g(s)\phi(t,s)dsdt=\int_Rf(t)g(t)\rho(t)dt $$ of course both integrals are supposed ...
user1326164's user avatar
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1 answer
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Calculate $\sum\limits_{n=0}^{\infty}{\int\limits_{\frac{1}{2}}^{\infty}(1-e^{-t})^{n}e^{-t^2}dt}$

$$ \mbox{Calculate}\quad \sum_{n = 0}^{\infty}\int_{1/2}^{\infty} \left(1 - {\rm e}^{-t}\right)^{n}{\rm e}^{-t^{2}}{\rm d}t $$ Basically I don't know where to start. I was thinking of using ...
Ranko's user avatar
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1 vote
2 answers
76 views

Is there a closed form for the integral $\int_{0}^{\pi/4}x^{k}\ln\tan x\, dx$, where $k$ is a natural number?

To start, I am aware that our integral $I(k)=\int_{0}^{\pi/4}x^{k}\ln\tan x\, dx$ is equal to $$I(k)=\int_{0}^{\pi/4}x^{k}\ln\sin x\, dx-\int_{0}^{\pi/4}x^{k}\ln\cos x\, dx$$, but I cannot seem to ...
Kisaragi Ayami's user avatar
1 vote
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13 views

Definition of integral of an almost everywhere non-negative, measurable function

Let $(X, \mathcal{M}, \mu)$ be a measurable space and $(X, \overline{\mathcal{M}}, \overline{\mu})$ be its completion. Let $f \colon X \longrightarrow \overline{\mathbb{R}}$ be an almost everywhere ...
user1063822's user avatar
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14 views

Integral by parts on a scalar field over a curve

Let $M$ be a compact and boundaryless Riemannian manifold. Take $f\in C^\infty(M)$ and let $T_s(x)=\exp_x(s\nabla f(x))$ be its gradient flow. I have proven for my specific case that $$\int_0^1\Delta ...
Gomes93's user avatar
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3 answers
72 views

Proving that $\iint f(x) g(x+y) dx dy \leq \int g^2(x) dx$

Let $D \subset \mathbb{R}^n$ be bounded, and $f,g: \mathbb{R}^n \rightarrow \mathbb{R}$. Furthermore let $f$ be nonnegative and such that $\int_D f dx= 1$. I would like to prove that $$\int_D \int_D f(...
CBBAM's user avatar
  • 6,073
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0 answers
30 views

Why is this proof about integration correct? [duplicate]

I have already asked about this particular integral, but I am not sure if this reasoning makes sense. From the equality $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-x^2-y^2}\ dxdy=\int_{-\infty}...
MSU's user avatar
  • 185
3 votes
0 answers
60 views

Help with Integral involving Square-root of Hyperbolic functions (definite)

I'm trying to solve the following integral for a project that I'm doing \begin{equation} I = \int_{x_0}^{x_1} \sqrt{\frac{a^2}{\sinh^2 x}-\left(b - \frac{c}{\tanh x} \right)^2} \, dx \end{equation} ...
MultipleSearchingUnity's user avatar
-1 votes
2 answers
59 views

Upper bound on a definite integral [closed]

This is from a Courant Comprehensive Exam in the $70$'s: $\mbox{Prove that the integral}\int_{3}^{5}\sqrt{x^{2} + 4}\,\,{\rm d}x < 10.$ Please solve this.
Callie12's user avatar
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1 answer
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Is there a function $\phi$ such that $\int_0^t\int_0^t \phi(u,v) dudv=\int_0^t\psi(s)ds$?

Let $t$ be a positive real number and $\psi$ a positive function. I am looking for a function $\phi$ such that: $$\int_0^t \int_0^t \phi(u,v) du\,dv=\int_0^t\psi(s)ds$$ Does such a function exist or ...
user1326164's user avatar
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Is it able to get the value of the intersection point between this two graph?

I'm the student who lives in Korea. I hope you guys understand my poor english level. So, this is my question: Does anyone know how to get the value of the intersection point between the two ...
mangotangogo's user avatar
2 votes
1 answer
47 views

Solving $\int\frac{1+\ln x}{3+x\ln x}dx$

I try to solve $\int\frac{1+\ln x}{3+x\ln x}dx$ in this manner: $ \int\frac{1+\ln x}{3+x\ln x}dx = \int\frac{1}{3+x\ln x}dx + \int\frac{\ln x}{3+x\ln x}dx $ Then we know answer of $\int\frac{\ln x}{3+...
alirezaarzehgar's user avatar
3 votes
1 answer
36 views

Generating functions of Wallis integrals

simple question, how would one go about proving that $$ \sum_{n\ge 0}^{}W_{n}x^{n}=\int_{0}^{\frac{\pi}{2}}\frac{dt}{1-x\cos t} $$ from this, it results with substituing $u= \tan(t/2)$ that $$ \sum_{n\...
Skepta's user avatar
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-1 votes
1 answer
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Looking for a specific function with integration

let t be a positive real number $t>0$. I'm looking for a function $\phi$ (independent of $t$) such that $$\int_0^t \int_0^t \phi(u,v)dudv=t$$ Can anyone told me if such a function exist or not ?
user1326164's user avatar
2 votes
4 answers
69 views

Evaluate $\lim_{n \to \infty} \int_0^\infty \frac{1+ \frac{x}{\sqrt{n}} e^{-x/n}}{(x+1)^2} \hspace{0.1cm} dx$

Evaluate $$\lim_{n \to \infty} \int_0^\infty \frac{1+ \frac{x}{\sqrt{n}} e^{-x/n}}{(x+1)^2} \hspace{0.1cm} dx$$ As this is a limit of integrals, surely we will be using a (monotone/dominated/bounded) ...
Grigor Hakobyan's user avatar
1 vote
1 answer
79 views

$\lim_{n\to+\infty}\int_{0}^{2\pi}\left(1+\frac{\sin(x)}{n}\right)^{\frac{n}{x}}\,\mbox{d}x$

I've been trying to compute the following limit: $$ \lim_{n\ \to\ \infty}\int_{0}^{2\pi} \left[1 + \frac{\sin\left(x\right)}{n}\right]^{n/x} \,{\rm d}x $$ I am not completely sure, but when $x\in (0,...
Giovanni Petrone's user avatar
3 votes
4 answers
558 views

Why Learn Measure Theory and Lebesgue Integration?

As someone who has taken two semesters of real analysis, having been exposed to the rigorous definition of the Riemann-Stieltjes integral - why should I learn Lebesgue integration? The Riemann ...
zaccandels's user avatar
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4 answers
102 views

Evaluate $\int {\frac{\sin {8x}}{\sin^4{x}+\cos^4{x}}}dx$ [closed]

I have solved 3 similar questions before where instead of $\sin{8x}$ there were $1$, $\sin{x}$, $\sin{2x}$ and they were pretty easy but using double angle formula in this one gave me a numerator of $$...
Mahi Alam's user avatar
1 vote
2 answers
87 views

Any method to evaluate or prove the divergence of $\int_0^{\infty} \frac{x^n \tan ^{-1} x}{1+x^2+x^4} d x$?

Recently, I met a difficult integral in the post $$ \int_0^{\infty} \frac{x^2 \tan ^{-1} x}{1+x^2+x^4} d x = \frac{\pi^2}{8 \sqrt{3}}+\frac{\pi}{24} \ln \left(\frac{2-\sqrt{3}}{2+\sqrt{2}}\right)+\...
Lai's user avatar
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1 vote
0 answers
27 views

Using Leibniz Integral Rule to Evaluate a log-sine Integral

I am looking for someone to check my work here. $$I(a)=\int_0^{\frac{\pi}{2}}\ln(1+a\sin^2(x))dx$$ Using Leibniz/Feynman's trick, I differentiate both sides with respect to $a$, leaving me with $$I'(a)...
Grey's user avatar
  • 373
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0 answers
16 views

Volume of shifted and modified cylinder using polar coordinates

I have a shifted and modified cylinder $x^2+y^2=4x$, bounded below by $z=0$ and above by $z=\sqrt{16-x^2-y^2}$. I want to find its volume. Completing the square and conversion to polar coordinates ...
mohd's user avatar
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3 votes
1 answer
95 views

Solving $\int\sqrt{x+\sqrt{x+\sqrt{x+\dots}}}\ dx$

I have attempted below, but I would appreciate any corrections or even a different, more efficient way to solve it. $$I = \int\sqrt{x+\sqrt{x+\sqrt{x+\dots}}}\ dx$$ consider: $y=\sqrt{x+\sqrt{x+\sqrt{...
Grey's user avatar
  • 373
0 votes
1 answer
40 views

Working on details on the Secretary Problem

I've been trying to follow this proof of the optimal way to solve the secretary problem (ref. https://en.wikipedia.org/wiki/Secretary_problem). Everything is clear to me except where they are ...
Alex's user avatar
  • 134
-3 votes
0 answers
53 views

Does square-integrability imply quartic-integrability? [closed]

Consider a smooth function $f$ that is square integrable, i.e. $$\int_{\mathbb{R}} dx \, f^2(x) < \infty$$ Does square integrability of $f$ imply integrability of some higher powers of $f$, e.g. ...
Octavius's user avatar
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2 votes
1 answer
50 views

Is there a short proof of the Second Mean Value Theorm for Integrals (strong, preferably asymmetric version)

The parenthesis in the title comes from the fact that there are essentially six versions of the conclusion in what may be called 2nd Mean Value Th. for Int. - not including special variants like those ...
Ulysse Keller's user avatar
0 votes
1 answer
39 views

Contour integration and how direction effects their bounds

I have started complex analysis and have been introduced to the idea of integrating in a positive direction for a simple closed curve. I was asked to show that $\int{e^z} dz$ = 0 along the contour ...
Oofy2000's user avatar
  • 598
2 votes
1 answer
92 views

Integration of an Equation that I'm unable to solve [closed]

There's this equation I'm unable to integrate, and I'm good at integration, but only the ones which are not very complex because I've just started learning integration this year. I need help to find ...
Moon_Hawk77's user avatar
3 votes
2 answers
58 views

Upper bound of polynomial integral

In my recent research, I encountered the following integral inequality $$\int_0^1kx^{k-1}(1+x)^{2k+1}\mathrm{d}x<2^{2k},$$ where $k$ is a positive integer. This inequality can be transformed to ...
Rookie's user avatar
  • 98
1 vote
0 answers
18 views

How can we fill the integration bounds and define a function $s$ from the given data so that this double integral is the mean of this indefinite int?

Suppose $f:\mathbb R \to \mathbb R$ has $\int_{-\infty}^\infty f(t)\ dt = 1 \in \mathbb R$. Define $g : \mathbb R \to \mathbb R$ by $g(\alpha) = \int_{-\infty}^\alpha f(t)\ dt$ and define $h : \mathbb ...
Snared's user avatar
  • 904
1 vote
1 answer
33 views

Trouble with 3D Fourier transform of a cross product expression

I don't understand a Fourier transform identity that has been quoted with no source in several papers (relevant to quantum mechanics): $$ \vec f(\vec r_i)=\int \frac{\vec r_i - \vec r_j}{|\vec r_i - \...
user2188518's user avatar
0 votes
0 answers
27 views

Smoothness of expectation of a piecewise function

Suppose $f(x)$ and $g(x)$ are piecewise smooth functions. For simplicity, we can assume that $f(x)$ has $m$ pieces, and $g(x):=\max_{i=1,2,\ldots, I}\left\{k_i~ x+b_i\right\}$. I have two questions: ...
Vergil's user avatar
  • 97
0 votes
0 answers
32 views

Change order of Limit and Integral pitfalls?

Probably it is a silly question. Say, we are given a function $$ Y(\epsilon)=e^{\epsilon z}\left(\frac{z}{\epsilon}-\frac{1}{\epsilon^2}\right) $$ to evaluate its limit as $\epsilon\to0$. I ...
MathArt's user avatar
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0 votes
1 answer
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Integral of $\frac{e^{iz}}{(z^2+a^2)(z-i)^2}$ on the rectangle with corners $z=\pm 1$ and $z=\pm 1 + (1+a)i$, negative way

Note that $$ \int_{C-}\frac{{\rm e}^{{\rm i}z}}{\left(z^{2} + a^{2}\right)\left(z - {\rm i}\right)^{2}}{\rm d}z = -\int_{C}\frac{{\rm e}^{{\rm i}z}} {\left(z^{2} + a^{2}\right)/\left(z - {\rm i}\right)...
Wrloord's user avatar
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0 votes
1 answer
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A 4d integral, Exercise I-8 p.68-69 from "Mathematics for the physical sciences" (Dover), Laurent Schwartz

The first part is about two versions of a formula by Feynman for the inverse of a product, one of which being $$ \frac{1}{a_1\, a_2 \, \cdots\, a_n} = \int_0^1 \int_0^{1-x_1} \cdots \int_0^{1-x_1 -x_2 ...
Noix07's user avatar
  • 3,624
1 vote
1 answer
95 views

Evaluate $\int_{0}^{\frac{\pi}{2}}\frac{\sin x}{(\sin^2 x-16)(\sin x-3)}dx$

I first tried to integrate using Integration by partial fraction. Then I got $\frac{\sin x}{(\sin^2 x-16)(\sin x-3)}=\frac{1}{2(u-4)}-\frac{1}{14(u+4)}-\frac{3}{7(u-3)}$ where $u=\sin x$ but don't ...
Mahi Alam's user avatar
0 votes
1 answer
50 views

What is the exact reasoning behind integrating both sides of an equation with bounds?

Say I have an equation: $$\frac{dy}{dx}=2y$$ where $y = 1$ when $x = 0$. Rearrange: $$dy=2ydx$$ Integrate: $$\int^y_1 dy=\int^x_0 2ydx$$ My question is why can we do this last step. I don't see the ...
BadUsername's user avatar
-1 votes
2 answers
113 views

Evaluate integral $\int_0^1 \frac{\ln(1-x)+x}{x^2}~dx$ [closed]

Evaluate : $$\int_0^1 \frac{\ln(1-x)+x}{x^2}~dx$$ I had the idea of using convergence, split it in two integrals, but I can't end it.
Pics Deb's user avatar
0 votes
1 answer
73 views

Trouble calculating an integral in the complex plane

The integral in question is $$2 \int_0^{+\infty}z^{2b-1} sinz \cdot dz \quad \quad b>0$$ I have tried 2 different but similar approaches, one calculating the integral $\int_{-\infty}^{+\infty}z^{...
0 votes
0 answers
53 views

A question about how substitution methods work in integration

In trigonometric substitution, it's common to say $t=\tan(x/2)$ for integrals involving $R(\sin x,\cos x)$. However, this is also done when $\tan(x/2)$ can be undefined for values where the integrand ...
illtry's user avatar
  • 1
1 vote
3 answers
89 views

Convergence of $\int_{0}^{+\infty} \frac{\log(x)}{x^2-1}dx$

I'm trying to solve the following complex analysis problem: Show that for all $n>1$, the integral $\int_{0}^{+\infty} \frac{\log(x)}{x^n-1}dx$ converges and in that case compute the integral using ...
Giovanni Petrone's user avatar
0 votes
1 answer
18 views

Verification of a Transformed Random Variable

I am looking at the technique used to determine the pdf of a transformation of a random variable. In Example 22.1, given $f_X(x)=3x^2,Y=X^2$, they have calculated $$f_Y(y)=\sqrt{y}$$ which I ...
Starlight's user avatar
  • 1,774
1 vote
2 answers
106 views

How to evaluate $\int_{0}^{1} \int_{0}^{1} \frac{4y \, da \, dy}{(y^2 + ay + 1)(y^2 - ay + 1)}$? [duplicate]

Question: How to evaluate $$\int_{0}^{\pi/2} \ln \left( \frac{2 + \sin x}{2 - \sin x} \right) \, dx$$ My attempt The original integral is: $$ J = \int_{0}^{\pi/2} \ln \left( \frac{2 + \sin x}{2 - \...
Martin.s's user avatar
1 vote
0 answers
103 views

Help integrating $\int_{-1}^1 \frac{1}{1 + e^{x^2}} dx$ [closed]

I was wondering if I could get help with integrating $$\int_{-1}^1 \frac{1}{1 + e^{x^2}} dx$$ This integrals comes up in the context of partition functions in statistical mechanics for Heisenberg spin ...
amongus's user avatar
  • 29
-1 votes
0 answers
13 views

Suitable techniques for integrating moments of arbitrary kernels?

Hello calculus experts, I'm doing statistical analysis, and I want to find moments of different well-behaved functions. Most of them are symmetric, smooth, quickly convergent, etc. I want to see if I ...
Machinus's user avatar
-2 votes
0 answers
79 views

value $\sum^{2024}_{k=1013}\frac{1}{k}=$ [closed]

Is there a very simple method to calculate this summation? $\sum^{2024}_{k=1013}\frac{1}{k}=$
deval sidi's user avatar
0 votes
2 answers
67 views

Is there a closed form for the integral $\int_{0}^{\infty}\ln(z)z^{\lambda - 1}\exp\left(-\frac{w}{2}\left(z + \frac{1}{z}\right)\right)\mathrm{d}z$?

As the title says, I would like to know if there is a closed form for the integral: \begin{align*} \int_{0}^{\infty}\ln(z)z^{\lambda - 1}\exp\left(-\frac{w}{2}\left(z + \frac{1}{z}\right)\right)\...
learner123's user avatar
0 votes
1 answer
61 views

Riemann integrability for step function

Here is the problem: Fix $c\in\mathbb{R}$ and define $g:[0,2]\to\mathbb{R}$ by $$g(x)=\begin{cases}2 &\text{if } 0\le x<1\\c &\text{if } x=1\\ 1&\text{if } 1< x\le 2.\\\end{cases}$$ ...
Sym Sym's user avatar
2 votes
1 answer
57 views

Evaluation of the given line integral

Question: Evaluate $\int_{C}$B.dr along the curve $x^{2}$+$y^{2}$=1,$z$= 1 in the positive direction from (0,1,2) to (1,0,2);given B= (xz²+y)i+(z-y)j+(xy-z)k The question itself is easy,but I don't ...
The Sapient's user avatar
0 votes
1 answer
26 views

I am trying to show that a certain parameter-dependant function is continuous on R+

the function is $\Phi(x)=\int_{-\pi}^{\pi} e^{x f(t)} dt, \quad (x, t) \in \mathbb{R}^+ \times \mathbb{R}$ and $f(t)=e^{it}-1-it $ first i proved that: $|e^{xf(t)}|= e^{-2xsin(\frac{t}{2})^2}$ $|e^{...
Baka's user avatar
  • 3
0 votes
1 answer
50 views

The same equation giving different integrals?

I feel like I’m missing something obvious. I have checked on online integral calculators and I keep getting different answers despite the fact they are equivalent fractions. $$\frac{1}{0.5x+5}=\frac{2}...
arung's user avatar
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