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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12 views

How to compute the triple integral over this region?

The region K is defined by: $ 0 \leq y+x \leq 4-(x-y)^2 $ $ y^2+x^2 \leq z \leq 2(1+x+y) $ So far I could reform the terms and it looks something like this: $ 0 \leq y+x \leq ...
2
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2answers
25 views

When is it insufficient to treat the Dirac delta as an evaluation map?

The Dirac delta "function" is often introduced as a limit of normal distributions $$\delta_a(x)=\frac{1}{a\sqrt{\pi}}e^{-\frac{x^2}{a^2}}\text{ as }a\to\infty.$$ Obviously, this sequence of functions ...
-2
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0answers
33 views

Ziploc Conjecture

A flexible plastic bag width $w$ and height h is filled with a liquid of volume V almost fully to roundness, sealed at top and placed on a flat table with its height approximately vertical. Prove ...
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0answers
31 views

Are there functions for which the cyclic integration-by-parts technique does not work?

There are a lot of functions where you can use what my teacher has described as the 'cyclic' method of integration. An example is $$\int e^x\sin x\,dx$$ where you designate $u=\sin x$ and ...
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1answer
22 views

Lp spaces are nested but then why is 1/x square summable but not summable?

If $1\leq s<r<\infty$ and $f\in L^r$ then $f\in L^s$, so then why is $\frac{1}{x}$ not in $L^1$ but is in $L^2$ for the counting measure $c:\mathbb{N}\rightarrow \mathbb{R}$?
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2answers
27 views

How to get the radii for this volume integral?

I need to find the volume of the solid obtained by rotating the region bounded by the curves $y=x, y=0, x=2$, and $x=4$ around the line $x=1$ I know I need to integrate $\pi*((\text{outer radius})^2 ...
4
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1answer
27 views

Is there a simpler approach to this application of Dominated Convergence?

For a measure theory class, I'm trying to evaluate: $$\lim_{n\to\infty}\int^\infty_1\frac 1 {nx} e^{-x/n}\ \text d\lambda$$ Obviously I want to try and move the limit through the integral and ...
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1answer
19 views

Limit of a summation, using integrals method

I have seen an interesting question on stackexchange, which I would like to requote so that I can understand the answer =) $\lim\limits_{n\to\infty} \dfrac{1^{99} + 2^{99} + \cdots + ...
1
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1answer
26 views

Change of integrals

This is probably ridiculously easy, but I can't figure this out: Why is $$\int_0^\infty \frac{e^{-t}}t \int_{|x-y|^2/(2t)}^\infty e^{-s} \,ds\,dt =\int_0^\infty e^{-s} \int_{|x-y|^2/(2s)}^\infty ...
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0answers
8 views

proof of a special case of discrete-time tower property

I'm reading a book on stochastic process and the first chapter is about properties of conditional expectation. One of the example the book gives is the proof of a special case of tower property in ...
3
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1answer
18 views

Estimate the Cauchy integral for matrix-valued functions

Recently, I became familiar with the concept of the "matrix function via Cauchy integral", i.e., $$f(A):=\frac{1}{2\pi i}\int_\varGamma f(z)(zI-A)^{-1} \mathrm{d}z$$ Furthermore, it can be shown that ...
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1answer
27 views

dominated convergence theorem and solve an integral [on hold]

I will solve this integral from dominated convergence theorem can you show me how: $$\int^1_0\frac{t^{a-1}}{1+t^{b}}dt=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{nb+a}$$
2
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0answers
79 views

To determine a definite integral

I have been trying to solve the following integral $$\int_{0}^{\frac {\pi}{2}} \ln\left (\frac {\ln^2 (\sin x)}{\pi^2+\ln^2 (\sin x)}\right) \frac {\ln \cos x}{\tan x} dx$$ I tried substituting for ...
2
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2answers
25 views

When to use Integral Substitution?

$e^x$$(1+e^x)^{1\over{2}}$ why can't use integration by part, What is meant by in the form of f(g(x))g'(x)? Can you give a few example? Thank you
2
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1answer
45 views

Double integral polar coordinates for area bounded by curves

I need to find the area bounded by the curves : $$ x^2 + y^2 = 1, \ y^2= x\sqrt3, \ x \geqslant \frac{y^2}{\sqrt3} $$ My attempt: $$ \int^{\frac{\pi}{2}}_{\frac{3\pi}{2}} \int^{\text{some ...
5
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2answers
45 views

How to evaluate $\int_0^{\infty} \bigg(\frac{e^{-x}}{\sinh(x)} - \frac{e^{-3x}}{x}\bigg) \; dx$

Evaluate the integral below $$\int_0^{\infty} \bigg(\frac{e^{-x}}{\sinh(x)} - \frac{e^{-3x}}{x}\bigg) \; dx$$ Using Wolfram I get the integral is $\gamma + \log\bigg(\frac{3}{2}\bigg)$, where ...
2
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0answers
12 views

Fluid Flow: lubrication, integration, ODE

Basically, I'm modelling the flow of a "coating" process -- a fluid flow between a flat moving plane and a stationary cylinder, 2D, cartesian coordinates. Subscript 0 is the at the minimum height b/w ...
1
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1answer
24 views

Fourier transform (properties)

I have a function $f$ such that $|f(x)|\leq e^{-x^2/2}$ hence in $\mathcal{L}^2(\mathbb{R})\cap\mathcal{L}^1(\mathbb{R})$ and thus we can compute the Fourier transform $$\hat{f} (\xi) = ...
3
votes
1answer
15 views

Equal integrals, circles, opposite directions

I've found this equality in my complex analysis book, but I don't see why it is true. Could you help me understand it? $$\int _{\partial D(1,1)} \frac{dz}{(z-1)(z+1)} = \int _{- \partial D(-1,1)} ...
1
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1answer
18 views

Finite parameter integral implies finite norm

Need a bit of help with a parameter integral problem. We have, $X$ is a finite measure space with measure $\mu$ and $f:X\rightarrow [0 , \infty)$ is a measurable function. The parameter integral ...
-1
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1answer
39 views

Given $f(x)=x+\int_{0}^1 t(t+x)f(t) dt $ , what is $f(0) $? [on hold]

Let $f:\mathbb R \to \mathbb R$ be such that $f(x)=x+\int_{0}^1 t(t+x)f(t) dt $ , then how do we find $f(0) $ ?
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0answers
14 views

Evaluate the line integral with Euler.

Need some help evaluating this line Intergral. $\int$$_c$ xy${e^y}$$^z$ dy Where C: x = 4t ; y = 3t$^2$ ; z = 3t$^3$ ; 0$\le$t$\le$1 Any help would be great. Thanks.
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0answers
25 views

Why does this integration proof not work for proving the formula for the surface area of a cone?

I know there is a similar question to this one. However, the method used there is slightly different; I would like to know what is wrong with my method I tried to prove that the surface area of a ...
4
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1answer
53 views
6
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0answers
41 views

closed form for $I=\int_{0}^{\infty}\frac{x^n}{x^2+u^2}\tanh(x) dx$

solve $$I=\int_{0}^{\infty}\frac{x^n}{x^2+u^2}\tanh(x) dx:0<n<2$$ I tried for $n=1$ : $$I(v)=\int_{0}^{\infty}\frac{x}{x^2+u^2}\tanh(vx) dx$$ ...
1
vote
1answer
36 views

Proving that $F(x)$ is a constant

This was on a test and i know i was supposed to use 2nd ftoc to prove that $F(x)$ was a constant when $x>0$ $$ F(x) = \int_{0}^{x} \frac{1}{t^2 +1} dt + \int_{0}^{\frac{1}{x}} \frac{1}{t^2 +1} ...
0
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0answers
35 views

Are there ever cases where it's easy to get coefficients for the series representation for an integrand, but hard to approximate the integral?

WHY I'M ASKING THIS I'm working on a faster way to approximate integrals of series. So I'd like to know if this could be useful. THE QUESTION If we suppose that we can get a formula for the ...
7
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2answers
86 views

Prove that $\int_0^\infty \frac{\ln x}{x^n-1}\,dx = \left(\frac{\pi}{n\sin\left(\frac{\pi}{n}\right)}\right)^2$

This question inspired me to ask the following. Prove that $$I_n = \int_0^\infty \frac{\ln x}{x^n-1}\,dx = \left(\frac{\pi}{n\sin\left(\frac{\pi}{n}\right)}\right)^2,$$ for $\Re(n)>1$. For some ...
0
votes
2answers
36 views

Finding the volume of revolution using the method of shells

I'm trying to find the volume of the solid generated by revolving the region bounded by $y=x^2$ and $y=6x+7$ about $x$-axis using the shell method. I applied the method and I got $15864/5$ ...
1
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0answers
15 views

integration with pade approximant

given the function $$ \int _{0}^{\infty}\sqrt{x}exp(-x) $$ can we use Pade approximants to integrate this i mean let bhe te rational approxsiamtions of $ \sqrt{x}= \frac{A(x)}{B(x)} $ and $ ...
0
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0answers
7 views

Another mutivariable integral over a simplex

Let $p$ and $q$ be two positive integers and let $\beta \neq 1/2$ be a real number. Then let $B > A > 0$. With the help of Mathematica, ie by doing elementary integrations and by consecutively ...
4
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2answers
49 views

Solving Integrals w/Trig

I need to solve the following integral: $$\int \sin^2(x)\cos^2(x) dx$$ This problem belongs to math notes that can be found here. Here are the steps listed to solve the equation. I can solve to a ...
3
votes
1answer
83 views

Closed form of $I=\int_{0}^{\pi/2} \tan^{-1} \bigg( \frac{\cos(x)}{\sin(x) - 1 - \sqrt{2}} \bigg) \tan(x)\;dx$

Does the integral below have a closed-form: $$I=\int_{0}^{\pi/2} \tan^{-1} \bigg( \frac{\cos(x)}{\sin(x) - 1 - \sqrt{2}} \bigg) \tan(x)\;dx,$$ where $\tan^{-1} (\cdot)$ is inverse tangent function. ...
0
votes
1answer
6 views

How to find the limits of integration to get the area for a loop of a lemniscate?

I know how to integrate the squared radius to get the equation that'll give me the area, like such for a lemniscate with $r^2=8\sin(2\theta)$ : $$1/2\int 8sin(2\theta) = 4 \int \sin(2\theta) = 4 * ...
1
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2answers
46 views

What is $\int_0^{\infty} x^2e^{\frac{(x-\mu)^2}{2 a^2}} dx$?

How can we express the integral $\int_0^{\infty} x^2e^{-\frac{(x-\mu)^2}{2 a^2}} dx$ for example by means of the error function? The problem is of course, that the expectation value is shifted and we ...
1
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0answers
24 views

Can I do the following when solving my integration??

I appreciate any feedback for my question. I have an integration as follows $$\int_{-\pi}^{\pi}\frac{1}{2\pi} \prod_i \frac{1}{1+ x_ig(\theta)} d\theta $$ I have that $g(\theta)$ is the defined as ...
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0answers
410 views

Relation I found: $ (\sum_{r=1}^{\infty}\frac{z(r)}{r})\times \int_0^\infty f(x) dx = \lim_{h \rightarrow 0} \sum_{i=0}^{n} f(k_ih)h$

I was fiddling with some maths and came up with an interesting relationship: $$ (\sum_{r=1}^{\infty}\frac{z(r)}{r})\times \int_0^\infty f(x) dx = \lim_{h \rightarrow 0} \sum_{i=0}^{n} f(k_ih)h$$ ...
8
votes
2answers
107 views

Test for convergence $\int_0^{\infty} \frac{\sin(x)}{x+\log(x)} \ dx$

What is the easiest way to test the convergence of $$\int_0^{\infty} \frac{\sin(x)}{x+\log(x)} \ dx$$ Is it possible to only use the high school tools for that?
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5answers
58 views

Does $\int_0^{\infty}\frac{x\hspace{1mm}dx}{x^3+1}$ converge? [on hold]

Does $\int_0^{\infty}\dfrac{x\hspace{1mm}dx}{x^3+1}$ converge? Can some explain how to approach this problem? All ideas are appreciated
0
votes
1answer
25 views

what is the order of integration for : integral of x dx * integral of y dy

I'm still trying to get my head around he basics of this stuff so please use simple language in your answer ! $$ \int dx \int f(x,y) dy$$ the first integral limits are from 0 to 1 for dx and the ...
-1
votes
2answers
25 views

Integral Differentiation over constants [on hold]

Let $f(x)$ be integrable on $x\in[0,X]$ and $a,b>0$ constants. I would like to get the derivative of $$I(x)=a\int^x_0{(b-X-f(x))dx}$$ with respect to $b$, i.e. $\dfrac{\partial I(x)}{\partial b}$. ...
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0answers
35 views

Is it possible to solve this set of equations?

Let's have system of equations: $$ \tag 1 [\nabla \times \mathbf E ] = -\frac{\partial \mathbf B}{\partial t} , $$ $$ \tag 2 [\nabla \times \mathbf B] = \sigma \mathbf E + A(\mu \mathbf K + C \mathbf ...
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0answers
32 views

How to solve the following integral [on hold]

Do you have any idea how to solve the following integral: $$\int\limits_0^a {{e^{\large \left(- \frac{{by}}{{c - dy}} - ey\right)}}dy}$$ where $a$, $b$, $c$, $d$ and $e$ are constants? Thank you ...
12
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5answers
167 views

Evaluation of $\int_0^{\pi/4} \sqrt{\tan x} \sqrt{1-\tan x}\,\,dx$

How to evaluate the following integral $$\int_0^{\pi/4} \sqrt{\tan x} \sqrt{1-\tan x}\,\,dx$$ It looks like beta function but Wolfram Alpha cannot evaluate it. So, I computed the numerical value of ...
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0answers
40 views

A question about $f(x)\equiv C$

Let $f(x)$ is Continuous function on $[0,\pi]$,and for $n=1,2,.....,$ the function $f(x)$ has the following property:$$\int_{0}^{\pi}f(x)\cos{(nx)}dx=0.(n=1,2,......)$$ Proof: $f(x)\equiv C$(C is ...
0
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0answers
8 views

Integral over homogeneous function does not vanish

Let $\alpha>0$ be a multi-index. For $x,y\in\mathbb{R}^n$, $n>1$, we consider the integral $$\int_{|x|=1} \int_{|y|=1} \partial_y^\alpha f(y)\ g(x,y)\ \mathrm{d}y \mathrm{d}x\qquad (*)$$ where ...
2
votes
2answers
95 views

How to show that $\int_0^1 dx \frac{1+x^a}{(1+x)^{a+2}} = \frac{1}{a+1}$?

From numerical evidence it appears that whenever the integral converges, $$J_a :=\int_0^1 dx \frac{1+x^a}{(1+x)^{a+2}} = \frac{1}{a+1}.$$ For $a \in \mathbb{N}$, I was able to prove this using ...
1
vote
0answers
22 views

Accelerometer data integration (MMSE)

Based on the raw accelerometer measurements use simple integration on the raw $X$ and $Y$ axis data to determine the velocity and position. If we assume a linear model $Y = aX + b$ for determine the ...
1
vote
2answers
36 views

Integral of pdf

I need to find the integral for this pdf but I don't know if I need to, or can, take the integral of two variables at the same time. $$ f(x;\theta)=\frac{x}{\theta^2} e^{-x^2/(2\theta^2)} ,\quad ...
2
votes
1answer
47 views

If functions converge a.e. and their integrals converge, does convergence in $L^1$ follow?

I was wondering if $f_n, f:\mathbb{R}\rightarrow\mathbb{R}$ are s.t. $f_n\rightarrow f$ pointwise a.e. and $\int f_n\rightarrow \int f$ where integrals are Lebesgue Integrals, is there any Theorem or ...