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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37 views

Integrate over components of the unit vector

if I have a vector field$X: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ and $\phi:\mathbb{R}^3 \rightarrow\mathbb{R}$ such that $X(r, \theta, \phi) := \phi(r,\theta,\phi) e_j(\theta,\phi)$, where $e_j$ is ...
1
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1answer
55 views

Show that $\int_{\mathbb R^n}e^{|x|^{-n}}dx=$ Volume of n-sphere

I'm preparing for a calculus exam, I'd like help in solving this question. Let $x \in \mathbb R^n$, $|x|={(x_1^2+x_2^2+...+x_n^2)^{\frac{1}{n}}}$, Show that $$\int_{\mathbb R^n} e^{|x|^{-n}}dx$$ is ...
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1answer
22 views

if $f$ is in weak $L^p$ and $\phi$ is $C_0^{1}$ then $f \ast \phi$ is in weak $L^p$

Okay, so I'd like to know if what I wrote in the title is true. Suppose that $f \in L^{p,\infty}(\mathbb{R}^n)$ (weak $L^p$ space) and $\phi \in C_0^1(\mathbb{R}^n)$ [or even $C_0^{\infty}$ if it ...
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3answers
75 views

Showing that $\frac{e^x + e^{-x}}{e^{2x} + e^{-2x}}$ is lebesgue integrable

I'm having some real trouble with lebesgue integration this evening and help is very much appreciated. I'm trying to show that $f(x) = \dfrac{e^x + e^{-x}}{e^{2x} + e^{-2x}}$ is integrable over ...
3
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1answer
85 views

Closed form of a trigonometric integral sought

I am trying to evaluate the definite integral $I(a,b)$, with $a,b\in\mathbb{R}$, defined by $$I(a,b):=\int_{0}^{2\pi}\sqrt{1-(a+b\cos{\theta})^2}\mathrm{d}\theta.$$ Assume $a,b$ are suitably ...
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1answer
21 views

Meaning of the unique up to a normalization factor

In the following text about Gaussian quadrature by Brian Bradie I cannot understand the meaning of the author: Associated with each weight function is a special family of polynomials, unique up ...
3
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3answers
216 views

A primitive function of $ e^{x^{2}} $

I made some efforts to set a closed form of primitive function of $ e^{x^{2}} $ i find this function : $ f(x)=\frac{x}{2x^{2}-1}e^{x^{2}} $ where : $f'(x)=(\frac{x}{2x^{2}-1}e^{x^{2}})'$= ...
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3answers
109 views

Evaluate $\int_0^{\infty}\int_0^{\infty}e^{-x^2-2xy-y^2}\,dx\,dy$

I would like to compute the following, $$ \int_0^{\infty}\int_0^{\infty}e^{-x^2-2xy-y^2}\ dx\,dy $$ It is obvious that we can rewrite the integral above to, $$ ...
1
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1answer
102 views

Evaluate $\int_{0}^{\infty}\sqrt{\frac{\sqrt{(a^2-y^2)^2+4y^2}+a^2-y^2}{(a^2-y^2)^2+4y^2}}dy=\sqrt{2}\pi$

Prove or disprove that$$\int_{0}^{\infty}\sqrt{\frac{\sqrt{(a^2-y^2)^2+4y^2}+a^2-y^2}{(a^2-y^2)^2+4y^2}}dy=\sqrt{2}\pi$$ for any $a>1$. I came across with this integral evaluating inverse ...
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0answers
41 views

Prove Green's theorem for circles

So the problem is in the title. The rules are that I can't split the circle into "rectangles" and I can't use pull-back. I tried to do something similar to the proof on unit squares. The problem is ...
2
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1answer
48 views

Showing function is lebesgue integrable

I have a function $f(x) = \frac{\sin(\frac{1}{x})}{1+(\log(x))^2}$ and I am trying to find whether this is Lebesgue integrable on $[1,\infty)$. I'm really not sure where to start on this one. It ...
3
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2answers
66 views

Evaluate the integrals $\int \sin{x} \cot^2{x} \,dx$ and $\int \cos{x} \cot^2{x} \,dx$.

Can you please show how to evaluate the integrals $$\int \sin{x} \cot^2{x} \,dx$$ and $$\int \cos{x} \cot^2{x} \,dx.$$
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2answers
73 views

Definite Integration: $\;36\pi \int_{0}^{5}t^3\sqrt{t^2+1}dt$

$\displaystyle 36\pi \int_{0}^{5}t^3\sqrt{t^2+1}dt$ I'm in Single Variable Calculus. This problem is about finding a surface area of parametric equations, and I have substituted values in. But I ...
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3answers
42 views

prove that $\frac{1}{h}\int^{h}_{0}F(y)\,dy\to F(+\infty)$ as $h\to +\infty$

If F is a non-decreasing, right continuous and a bounded function show that $$\frac{1}{h}\int^{h}_{0}F(y)\,dy\to F(+\infty)\text{ as }h\to +\infty.$$
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1answer
37 views

Euclidean space and vector field

Can someone explain me what a Euclidean space is? and more detailed what a vector field is? Or a continuous vector field
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0answers
38 views

If $f(x)$ is integrable on $[a,b]$ then $c\cdot f(x)$ is also integrable and $\int_a^b c\cdot f(x) dx=c\cdot \int_a^b f(x) dx$

I proved the first part of this theorem which says that $c\cdot f(x)$ is integrable,but how to prove that $\int_a^b c\cdot f(x) dx=c\cdot \int_a^b f(x) dx$? Maybe it provides a bit help if i tell how ...
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1answer
30 views

Integrating the error function in a calculation related to Brownian motion

I wish to calculate the probability that a standard linear Brownian motion $B(t)$, $t\ge 0$, will be at time $t_0$ inside the interval $[a,b]$, and at time $t_1$ in the interval $[c,\infty)$. To do ...
1
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1answer
27 views

Understanding claim in Newman and Barkema's Monte Carlo book

In Newman and Barkema's Monte Carlo Methods in mathematical physics, on page 23-24, the following claim is made: "Assume we have a function f(x) and the integral $I(x)=\int_0^xf(x')dx'$. Then pick a ...
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0answers
27 views

How to find the CDF of distance between two point in two circles respectively?

Let $C_1$ and $C_2$ be two circles with radius $R$ and $r$, $H$ be the distance between two centers, $H\in[0,+\infty)$, pick up Point $P_1(x,y)$ from $C_1$ and Point $P_2(a,b)$ from $C_2$ uniformly, I ...
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1answer
38 views

Why is the integral of the arc length in polar form not similar to the length of the arc of a circular sector?

So I learned that the area enclosed by a polar function is computed by $$A = \int \frac{r(\theta)^2}{2}d\theta.$$ Which, I learned, comes somewhat from the formula for the area of a circular sector ...
3
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2answers
83 views

$\int _0^{\pi }\:\sum _{n=0}^{\infty \:}\frac{n\cdot \sin \left(nx\right)}{e^n}dx=\frac{2e}{e^2-1}$

It is asked to prove: $\int _0^{\pi }\:\sum _{n=0}^{\infty \:}\frac{n\cdot \sin \left(nx\right)}{e^n}dx=\frac{2e}{e^2-1}$ I have tried to search for convergence and it gave me 0 so i can't solve it. ...
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3answers
91 views

Calculus: improper integration

If $\displaystyle L = \int_0^1 \dfrac{dx}{(1+x^8)} $, then what is the upper bound and lower bound for $L$. I tried with numerical methods. But not got the answer
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1answer
38 views

Integrate $f(x,y)=\begin{cases}(x-1/2)^{-3} &\text{if}, 0<y<|x-1/2| \\ 0 &\text{else} \end{cases}$

$f(x,y)=\begin{cases}(x-1/2)^{-3} &\text{if}, 0<y<|x-1/2| \\ 0 &\text{else} \end{cases}$ I have to integrate it over $E=[0,1]\times[0,1]$ I can integrate the inner integral ...
2
votes
4answers
67 views

Find the value of $a$ if $\int_0^\infty \frac{2x}{a}e^{\Large\frac{-x^2}{a}}\ dx=1 $

Find the value of $a$ if $$\int_0^\infty \frac{2x}{a}e^{\Large\frac{-x^2}{a}}\ dx=1 $$ I tried to use integration by parts but I didn't get a good response.
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1answer
23 views

Example of of sequence of continous functions

Give me example of sequence of continous real valued function defined on $[0,\infty)$. Suppose $f_n(x) \rightarrow f(x)$ for all x $\in [0,\infty)$ such that If $f_n \rightarrow f$ uniformly on ...
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0answers
43 views

pdf of area of a circle

$X,Y$ are random variables with standard normal distribution (they are independent). $W$ is the area of the circle that has center at $(0,0)$ and passes through $(X,Y)$. What is the pdf of $W$? I ...
2
votes
2answers
58 views

A Contour Integral I

What is the value of the integral \begin{align} \int_{-a}^{c} \sqrt{ \frac{a+x}{c-x} } \ \frac{dx}{(d-x)(x-b)} \end{align}
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1answer
101 views

Quick Question Integration with Joint PDF

Let $X_1, X_2, \ldots, X_n$ by independent and identically distributed random variables with probability density function (pdf) $$f_X(x) = \left\{\begin{array}{ll}1, & 0 < x < 1\\ 0, ...
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4answers
64 views

Is there a simpler way to compute this antiderivative?

I'm trying to compute the antiderivative $$\int \frac{y^2}{\sqrt{r^2 - y^2}} \, dy.$$ It is proving fairly tricky (for me). Here is Wolfram|Alpha's solution: $$\int \frac{y^2}{\sqrt{r^2 - y^2}} \, dy ...
5
votes
2answers
114 views

Why does this $u$-substitution zero out my integral?

Here's how I understand $u$-substitution working for an integral. Essentially, it involves substitution of differential expressions, allowing you to cancel out terms of the integrand. When we change ...
5
votes
3answers
258 views

General condition that Riemann and Lebesgue integrals are the same

I'd like to know that when Riemann integral and Lebesgue integral are the same in general. I know that a bounded Riemann integrable function on a closed interval is Lebesgue integrable and two ...
8
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2answers
166 views

Superelliptic Area Of $x^5+y^5=r^5$

$${\LARGE\int}_0^\tfrac\pi2\frac{dx}{\bigg(\sqrt[{\Large 5}]{\cos^5x+10\cos^3x\sin^2x+5\cos x\sin^4x}\bigg)^{\large 2}}~=~?$$ Its numerical value is about $1.40171345128228$. Maple, Mathematica, ...
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0answers
47 views

Functions which can not be integrated via Riemann.

I am looking for some (possibly exotic) functions which can not be integrated via Riemann integration but can be integrated via Lebesgue. I am aware of the rational indicator function, i.e., $$ f(x) = ...
4
votes
4answers
140 views

Question about $\frac{\sin(x)}{x}$ and $\frac{\cos(x)}{x}$

So here is my question. As known the famous integral $$ \int_0^{\infty} \frac{\sin(x)}{x}dx$$ converges an its value is $\frac{\pi}{2}$. As I was trying to solve a different integral today, after ...
2
votes
1answer
82 views

Spivak's “Calculus in Manifolds” problems

I have some troubles with this problems. Problem 1.18: If $A \subset [0,1]$ is the union of open intervals $(a_i,b_i)$ such that every rational number of $(0,1)$ is contained in $(a_i,b_i)$, for ...
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1answer
26 views

Would like to compute the limit of some integral sequence

Consider $$ \lim_{n\rightarrow \infty}\int_{\mathbb R}e^{-|x|n}e^{\frac{x^2}{2}}dx $$ The goal of an exercise I am working on is to compute the limit of the the integral above. By intuition it should ...
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1answer
45 views

Why is $\sum^{n}_{i=1} \int_{t_{i-1}}^{t_i} = \int_{a}^{b} $?

This is a part of my proof: $$\begin{align} \left| \sum^{n}_{i=1} V(r(\tau_i)) \cdot \int_{t_{i-1}}^{t_i} r'(t) dt - \int_{a}^{b} V(r(t)) \cdot r'(t) dt \right| &\leq \sum^{n}_{i=1} ...
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2answers
27 views

Exercise on abstract integration

Let $f_n$ be a sequence of nonnegative functions defined on $\mathbb{R}^N$ such that $f_n \rightarrow f $ almost everywhere on $\mathbb{R}^N$ and such that $$\int_{\mathbb{R}^N} f_n \rightarrow ...
1
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1answer
39 views

Integrate the differential form over a cardioid

$\omega=\dfrac{-ydx+(x-1)dy}{(x-1)^2+y^2}$ Calculate $\int_C\omega$ where $C...r=1+\cos\varphi$ (positively oriented) I'm still pretty lost when it comes to differential forms but as far as I ...
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1answer
44 views

Expanding the integrand gives a different result

I integrated this term in Mathematica: $$\int_{-\infty}^{\infty} d\omega\cdot \sin(s\cdot \omega)\cdot \frac{1}{e^{\beta\cdot \hbar\cdot \omega}-1}\cdot \frac{\omega}{\omega^{2}+\gamma^{2}}$$ The ...
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2answers
46 views

Computing integrals of differential forms

How do I compute the integral of the differntial form $\omega = xdy - y dx$ on $\mathbb R^2\backslash \{0\}$ along the path $\gamma(t) = (\cos(2\pi t),\sin(2\pi t))$? That is, what is ...
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1answer
35 views

Is there a statement of Fubini's theorem for real integrals making no reference to multi-variable integrals?

If $I$ and $J$ are two intervals of $\mathbb R$, then under what conditions can we say that $$\int_I \int_J f(x, y) dx\ dy=\int_J \int_I f(x, y) dy\ dx$$ Obviously we have to assume that $f$ is such ...
4
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5answers
200 views

$\int_0^{+\infty}{ \sin{(ax)} \sin{(bx)}}dx=?$

How can I calculate the integral: $$\int_0^{+\infty}{ \sin{(ax)} \sin{(bx)}}dx$$ ?? I got stuck.. :/ Could you give me some hint?? Do I have to use the following formula?? $\displaystyle{\sin{(A)} ...
0
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2answers
47 views

How to evaluate this triple integral?

How would I go about evaluating this integral? I want to change the order of integration but don't know how. $$\int_0^1\int_1^{\Large e^z}\int_0^{\log y}x\ dx\,dy\,dz$$ I'm having difficulty ...
2
votes
1answer
70 views

Compute an integral with residue theorem

Using residue theorem, compute the following integral: $$ \int_{0}^{2\pi}\frac{\left( 1+2\cos t\right) ^{n}\cos\left( nt\right) }{5+4\cos t}\operatorname*{dt}. $$ Or a source with a solution.
2
votes
3answers
82 views

Given that $\int_{0}^{100} (a^x-1) \:\mathrm{d}x = 30$, how can I calculate $a$?

How can I separate a from other constants so that I can evaluate a? I'm stuck at the bottom line as shown below. \begin{align} 30& =\int_{0}^{100} (a^x-1) \: \mathrm{d}x \\&= \left [ ...
0
votes
3answers
62 views

Evaluate integral

How do I evaluate the following integral, the answer according to Wolfram Alpha is $2$, but I keep on getting $0$ after using integration by parts.$$\frac12\int_{-\infty}^\infty x^2e^{-|x|}\ dx$$
1
vote
1answer
83 views

Does $\int_{-\infty}^{\infty}{\frac{\mathrm{exp}(-t^2)}{t-iz} dt}=i \sqrt{\pi} e^{z^2} \mathrm{erfc}(z)$ hold for all $z$?

I have been working on a calculation that involves the following type of integral: $$ f(z)={\frac{1}{i\sqrt{\pi}}}\int_{-\infty}^{\infty}{\frac{e^{-t^2}}{t-iz} dt} \hspace{1.5cm} z \in \Bbb{C} ...
0
votes
1answer
46 views

Calculating flux through a surface

Calculate the flux of the filed: $$F(x,y,z)=(3x-2y+z, 2x+3y-z, x-3y+z)$$ through the surface: $$S: |3x-2y+z|+|2x+3y-z|+|x-3y+z|=1$$ Any help would be much appreciated!
1
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3answers
103 views

Evaluate logrithmic integral

Does the following expression converge? Where $n$ is positive integer $1,2,3,...$ $$\int_0^\infty(\ln x)^n dx$$