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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9 views

An asymptotic expansion of $\int_{0}^{+\infty}\!\!\frac{dx}{\sinh^2(\epsilon \sqrt{x^2+1}) } $ for $\epsilon$ near $0$

How to find an asymptotic expansion, for $\epsilon$ near $0^+$, of the integral $$ I(\epsilon):=\int_{0}^{+\infty}\frac 1{\sinh^2 (\epsilon \sqrt{x^2+1}) } {\rm d}x. $$ As $\epsilon \rightarrow 0^+$, ...
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0answers
7 views

tetrahedron volume in the first octant

The surface is given: xyz = 2 It is in the first octant so x > 0, y > 0, z > 0. The tangent plane taken at any point of this surface binds with the coordinate axes to form a tetrahedron. Task: ...
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1answer
23 views

complex analysis fundental theorem of caculus

Can anyone please explain how $$\int \frac{1}{(z-2)^3}dz $$ evaluated about the closed continuous path $$1+3e^{i2t\pi}$$ is 0 by the fundamental theorem of calculus?
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1answer
22 views

Uniform continuity of the antiderivative

We know that if $f:\mathbb{R}\to\mathbb{R}$ is a function such that $$\sup_{x\in\mathbb{R}}|f(y)|<\infty,$$ then the function $g(x)=\int_0^xf(y)dy$ is uniformly continuous. I am just wondering ...
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0answers
20 views

Proving that $\frac{d}{dt}\int \Phi_t^*\omega=\int_{\Phi_t \circ \partial c} i_{\mathbb{X}}\omega$

Let $\omega$ be a closed $k$-form on $\mathbb{R}^n$ and $c:I^k \rightarrow \mathbb{R}^n$ a $k$-cube on $\mathbb{R}^n$. Let $\mathbb{X}$ be a vector field on $\mathbb{R}^n$ with flow $\Phi_t$. Show ...
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0answers
20 views

gamma functions and algebra.

Hi I am working through some algebra for something and I have copied the chunk I am stuck with $\int^\infty_y P(2y;2\nu,2k) dK$ where $P(2y;2\nu,2k)$ =$ \frac{1}{2} ...
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1answer
28 views

Evaluate Integral $\int_{c\ -\ j\infty}^{c\ +\ j\infty} \left({\sigma\,x^{-1}}\right)^s{\Gamma(\beta_1-1+s)\over \Gamma(\beta_1+\beta_2-1+s)}\,ds$

I am at this point of integration where: $$\int_{c\ -\ j\infty}^{c\ +\ j\infty} \left({\sigma\,x^{-1}}\right)^s{\Gamma(\beta_1-1+s)\over \Gamma(\beta_1+\beta_2-1+s)}\,ds$$ whereby $\beta_1$, ...
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1answer
40 views

Find the volume of the solid bounded by $ y=0 , z=1-x^2, z=x^2-1, y+z=1 $

i need to know the first steps and strategies of solving this kind of problems i know that i need to find the limits of x and y and then do the integral of a certain functions by dxdy
4
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2answers
30 views

how to integrate with square root with mode?

I want to integrate $$\int_{-1}^{2}\sqrt{|x|}dx$$ But I dont know how to do it should I integrate it fist from $-1$ to $0$ and then from $0$ to $2$ making one of equation in minus other in plus?
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1answer
64 views

Integration: definite integral from 0 to 1 [on hold]

How do I find the expression for the following: $$\int_0^1 x^n(1-x)^m \mathrm dx$$ for positive integers $n, m$?
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1answer
24 views

Prove that $\int_{X}\exp(f(x))d\mu(x)\cdot\int_{X}\exp(-f(x))d\mu(x)\geq\mu(X)^{2} $

Let $(X,\mathfrak{S},\mu)$ a measure space, and $f:X\rightarrow \mathbb{R}$ a measurable function. I have to prove that $$\int_{X}\exp(f(x))d\mu(x)\cdot\int_{X}\exp(-f(x))d\mu(x)\geq\mu(X)^{2}.$$ I ...
7
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1answer
57 views

How prove $ \int_0^1 f(x)dx-\exp\left(\int_0^1\log(f(x)) dx\right)\le\max_{0\le x,y\le 1}\left(\sqrt{f(x)}-\sqrt{f(y)}\right)^2 $

Consider a continuous function $f:[0,1]\to\mathbb{R}^{+}$. How show that $\int_0^1 f(x)dx-\exp\left(\int_0^1 \log(f(x)) dx\right)\le \max_{0\le x,y\le 1}\left(\sqrt{f(x)}-\sqrt{f(y)}\right)^2$?
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1answer
14 views

prove that $∀ε>0∃p∈P(U(f,p)−L(f,p)<ε)$

$F:[0,1]\times[0,1]\longrightarrow R$ $ f(x)= \begin{cases} 1, & \text{y<x} \\ 0, & \text{y $\geqslant$x} \end{cases} $ i have a problem choosing my p∈P and proving the statement any ...
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0answers
22 views

Cauchy's Residue Theorem for Integral $\int_{c\ -\ j\infty}^{c\ +\ j\infty} \left({\sigma \over x}\right)^s{{1-\beta^{s+1}}\over s(s+1)}\,ds$

I am at this point of integration where: $$\int_{c\ -\ j\infty}^{c\ +\ j\infty} \left({\sigma \over x}\right)^s{{1-\beta^{s+1}}\over s(s+1)}\,ds$$ where $\beta$, $\sigma$ and $x$ are real numbers ...
2
votes
2answers
49 views

How can I deduce the value of $\frac{1}{\sqrt{4\pi t}}\int_{-\infty}^{\infty}\sin(y)e^{-\frac{(x-y)^2}{4t} } dy$ without actually evaluating it?

How can I deduce that $$ \frac{1}{\sqrt{4\pi t}}\int_{-\infty}^{\infty}\sin(y)\,e^{-\frac{(x-y)^2}{4t} } dy = e^{-t} \sin(x) $$ without actually evaluating the definite integral?
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0answers
29 views

Evaluate Complex Integral with Term $\Gamma\left(s \over 2\,\right) \Gamma\left(\,{\beta +1 \over 2} - {s \over 2}\,\right)$

I am having trouble evaluating this integral: $$ \int_{c\ -\ j\infty}^{c\ +\ j\infty} \left(\,x^{-1}\sigma\,\right)^{s}\beta^{s \over 2}\ \Gamma\left(\,s \over 2\,\right) \Gamma\left(\,{\beta +1 ...
3
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1answer
64 views

Prove that $\int_0^{\infty} \int_0^{\infty} e^{-(x^2+y^2+2xy \cos \theta)} \,dx dy = \frac{\theta}{2\sin\theta}$

Prove that the following integral: $$\int_0^{\infty} \int_0^{\infty} e^{-(x^2+y^2+2xy \cos \theta)} \,dx dy = \frac{\theta}{2\sin\theta}$$ The hints written on the book are beta function and to ...
3
votes
1answer
35 views

What is the value of this integral (using the Argument Principle),

F(z) is given by $$F(z) = e^zz^{-2}(z-1)(z^2-4)(z+8)^7$$ What is the value of the integral $$\int_0^{2\pi} \frac{F'(3e^{i\theta})}{F(3e^{i\theta})}d\theta \space \space ?$$ I think the relevant ...
2
votes
2answers
57 views

Integrating $\int_{0}^{\infty}\frac{x^2{\rm d}x}{(x^2+9)(x^2+4)^2}$ [on hold]

Integrate $$I=\int_{0}^{\infty}\frac{x^2{\rm d}x}{(x^2+9)(x^2+4)^2}$$
1
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1answer
52 views

Is this a convolution?

I have the following integral \begin{align*} \int_{-\infty}^\infty f(t) q(t+ax) dt \end{align*} where a is some constant. This integral look a lot like convolution (or correlation). My question is ...
2
votes
1answer
37 views

Cauchy's Residue Theorem with Multiple Gamma Functions

I previously posted a similar problem here and here. This time however I am dealing with multiple gamma functions. This is the problem I am dealing with right now: $$ \int_{c\ -\ j\infty}^{c\ +\ ...
4
votes
2answers
74 views

Methods of evaluating $\int_0^{\infty}\frac{{\rm d}x}{x^2+1}$

Methods of evaluating $$\int_0^{\infty}\frac{{\rm d}x}{x^2+1}$$ Firstly i know that directly: $$\int_0^{\infty}\frac{{\rm d}x}{x^2+1}=\arctan x\Bigg|_{0}^{\infty}=\frac{\pi}2$$ Also we can use the ...
3
votes
2answers
45 views

How to use Stokes Theorem to evaluate $\int_{S} \text{curl} F\cdot d\mathbf{S}$

Let F = $( yz, 0, x)$ and $S$ is the portion of the plane ${x\over2} + {y\over3} + z = 1$ where $x, y, z \ge 0$, oriented with an upward pointing normal then prove: $$\int_{S} \text{curl} F\cdot ...
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2answers
35 views

General Solution for the Gravity Between Two 3D Triangles

I would like to find the general solution for the gravity between two (flat) triangles in 3D, including the location $(x,y,z)$ where this force should be applied (in order to later account for ...
2
votes
3answers
42 views

It takes 80J of work to stretch a spring 0.5m from its equilibrium position. How much work is needed to stretch it an additional .5m?

It takes $80\,\textrm{J}$ of work to stretch a spring $0.5\,\textrm{m}$ from its equilibrium position. How much work is needed to stretch it an additional $0.5\,\textrm{m}$? Here's what I have: ...
2
votes
1answer
23 views

Moment of inertia of a solid cone about a diameter of the base as axis

I have a solid cone base radius r, height h, mass M and I need to find the moment of inertia about a diameter of the base as axis. The book's answer is $\frac{1}{10}Mh^2+\frac{3}{20}Mr^2$ however, I ...
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votes
1answer
63 views

How to find the value of $\int_0^1 \frac{x-1}{\log x}\,dx$? [duplicate]

I want to find the value of $$\int_0^1 \frac{x-1}{\log x}\,dx$$ where $\log x$ stands for the natrual logarithm. I put it in the wolfram alpha, and it saids it's $\log 2$. (Refer to : ...
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1answer
32 views

$\int_{(0,\infty)}\frac{1}{x}\sin (\frac{\pi}{2}x) d\mu$ for certain measure $\mu$.

I have to check if $f(x):=\frac{1}{x}\sin (\frac{\pi}{2}x)$ is integrable in $(0,\infty)$ , with the measure $\mu$ where $\mu(A)=\mathrm{card}(A\cap\mathbb{N})$ if $A\cap\mathbb{N}$ is finite and ...
7
votes
6answers
184 views

Integration by differentiating under the integral sign $I = \int_0^1 \frac{\arctan x}{x+1} dx$

$$I = \int_0^1 \frac{\arctan x}{x+1} dx$$ I spend a lot of my time trying to solve this integral by differentiating under the integral sign, but I couldn't get something useful. I already tried: ...
0
votes
2answers
34 views

Integrating the absolute of the cosine

For some reason, I do not understand when computing the the integral of |cos(x)| from -pi to -pi/2 gives 1. When i compute it i get -1. There must be something I haven't understood.
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0answers
113 views

Integral. very nice [on hold]

Compute $$\int_0^1\int_0^1\int_0^1(x^2 + y^2 + z^2)\ln\left(\frac{1}{(x-a)^2 + (y-b)^2 +(z-c)^2}\right)\,dx\,dy\,dz.$$ $G = [0, 1]\times[0, 1]\times[0, 1]$, compute $$\int_{\partial ...
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votes
0answers
30 views

Definite integration $\int_{0}^\infty \frac{{x^2}}{e^{\beta {\big(\sqrt{x^2 + m^2}}- \nu\big)} + 1} dx$

$$I = \int_{0}^\infty \frac{{x^2}}{e^{\beta {\big(\sqrt{x^2 + m^2}}- \nu\big)} + 1} dx = ?$$ If a constant is added to the exponential in the denominator along with a square root in the exponent and ...
4
votes
3answers
215 views

Integration $\int_{m}^\infty {x}{\sqrt{x^2 - m^2}}e^{-\beta x} dx$

How can the following integration be performed? Does it involve Bessel functions?$$\int_{m}^\infty {x}{\sqrt{x^2 - m^2}}e^{-\beta x} dx$$ EDIT: Actually, the original question is: $$\int_{0}^\infty ...
1
vote
1answer
53 views

An integral with the $\Gamma$ function: $\int_{c- i\infty}^{c+i\infty} u^{s}\:\Gamma(\beta +s-1) \:ds$

I previously posted a similar problem here and I have solved many of the problems from the answers given with explanations. This time however I am at this point of integration where: $$\int_{c\ -\ ...
0
votes
0answers
31 views

Expectation of a discrete random variable

This might seem to be a dumb question but here it goes : I am trying to derive the expectation of a random variable by two different means. The first one : \begin{align} E(X) &= ...
0
votes
3answers
21 views

Integration substitution: How do you find the derivative of the denominator?

I have to integrate: $$ \int_1^2 \frac{37 x}{x^2-6 x+10} \, dx $$ $$ =37 \int_1^2 \frac{x}{x^2-6 x+10} \, dx $$ Then Wolfram Alpha tells me to rewrite the integrand as $$ \frac{2 x-6}{2 ...
1
vote
1answer
43 views

For a pdf $f(x)$, how can we prove that $\int_{-\infty}^{\infty} x\,f(x)\,dx=\int_{-\infty}^{\infty} F(x\geq t)\,dt$?

$f(x)$ is a probability density function and $F(x)$ is the corresponding cumulative distribution function, i.e., we have the relationship on the derivative $\frac{d}{dx}F(x)=f(x)$. Given this, how ...
1
vote
1answer
69 views

Evaluate $\int e^{\tan^{-1}x}(1+x+x^2)d(\cot^{-1}x)$

Integrate: $$\displaystyle \int e^{\tan^{-1}x}(1+x+x^2)d(\cot^{-1}x)$$ How to solve it? Integration with respect to $\cot^{-1}x$. I have started to change this integrand to integrate with respect ...
4
votes
0answers
73 views

Why do some authors write dx after integral sign? [duplicate]

Much has been said of the $dx$ notation used for integration on this site, but some writers of mathematics papers (especially physicists), write integrals as $$ \int dxf(x) $$ For instance, one way ...
1
vote
1answer
44 views

Integrate:$ \int (x^2+\cos^2 x)(\csc^2 x)/(1+x^2) dx$

Integrate: $[x^2+\cos^2x]\csc^2 x/(1+x^2) dx$ How to make the substitution here. I have tried to make the substitution as $\tan^{-1}x =t$ But got stuck further.
3
votes
2answers
87 views

Proof of a closed form of $\int_0^1(-\ln x)^ndx$

$$\int_0^1(-\ln x)^ndx$$ Is there a step-by-step solution to a closed form of this expression? I've tried using different representations to re-write the expression but I couldn't find anything I knew ...
0
votes
0answers
20 views

Limits of an integral expression (with CDF and PDF)

Consider the following integral expression: $$ I(a)=\frac{1}{1-F(a)}\int_a^b \! (b-x)f(x) \mathrm{d}x, $$ where $F$ is a continuously differentiable cumulative distribution function for a random ...
1
vote
1answer
26 views

Conditions on integration by parts with unbounded endpoint

I have the following theorem for integration by parts when both endpoints are finite: (Lebesgue integrals are used throughout) Let $a\le b$ be real numbers, and $f,g$ be functions continuous on ...
0
votes
1answer
16 views

Limits of an integration (triple integral) [on hold]

What is the limits of the integration of S, where S is the tetrahedron with vertices (0,0,0),(3,2,0) (0,3,0), (0,0,2)? In the process, we will obtain a function 2x + 6y + 9z = 18, then the limits can ...
0
votes
1answer
52 views

Is $\int_{0}^{2\pi} f(x)\, \text{e}^{-i\ell x} \, \text{d}x =0$?

Suppose $f,g \in \mathcal{L}^1(\mathbb{R} / 2 \pi)$ with $f(x)=g(mx),m \in \mathbb{Z}$. I want to show that $$ \forall \, \ell \in \mathbb{Z},\,\,\, \text{with} \, \ell \not\equiv 0 \, \text{mod} ...
3
votes
5answers
74 views

Find integral of absolute values by splitting integrals, $\int_{-1}^{4} (3-|2-x|)\, dx$

I have trouble splitting the integral $$\int_{-1}^{6} (5-|2-x|)\, dx$$ Tried so far: Split the 3 and the absolute value to two separate integrals. Draw absolute value graph. Integrate both. I ...
2
votes
1answer
33 views

Changing integral and summation

Let $$ f(x)=\left(\sum_{n=1}^\infty \frac{\cos(nx)}{2^n}\right)^2 $$ $$\hat{f}(n) = \frac{1}{2\pi}\int_0^{2\pi} \left(\sum_{n=1}^\infty \frac{\cos(nx)}{2^n}\right)^2 e^{-inx} dx$$ Easy to see the ...
1
vote
1answer
45 views

Explanation for summation complex analysis method

This is @Amad27 something happened to my account, which I will get fixed soon, so for now I will ask as a guest until the problem is fixed. Thanks. I saw this method of calculating: $$I = ...
0
votes
2answers
80 views

Resolution of limits like this: $\lim_{n\to+\infty}\int_{0}^{\frac{\pi}{2}} e^{-n \sin(x)} \,dx $ [on hold]

Good morning. Can you give an help to solve these limits? I have thought of using the uniform convergence $$a)\lim_{n\to+\infty}\int_{0}^{\frac{\pi}{2}} e^{-n \sin(x)} \,dx \\ ...
-1
votes
1answer
50 views

Evaluate the limit integral using the Lebesgue Dominated Convergence Theorem

I have tried to use the Lebesgue Dominated Convergence Theorem to evaluate: $$\lim_{n\rightarrow \infty} \int_{(0,1]} f_n \;d\mu $$ with $f_n(x)=\dfrac{n\sqrt{x}}{1+n^2x^2}$ and ...