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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26 views

Residue of a rational function

In this answer by Jack D'Aurizio, which is fantastic, I do understand that: If $f(z) = (\psi(-z) + \gamma)^2$ where $\psi(-z)$ is digamma, and $H_n$ (following) is harmonic number: $$\mathrm{Res} ...
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2answers
50 views

Integration: u-substitution and factoring out constants

I'm struggling solving the following indefinite integral with u-substition. Given: $\int 2\sqrt[]{2x-1} dx , u=2x-1$ I begin to solve by factoring out 2, thus: $2 \int \sqrt[]{2x-1} dx$ and ...
-1
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4answers
107 views

Compute $\int\frac{dx}{\sqrt{\tan x}}$ [closed]

Compute $\displaystyle \int\dfrac{dx}{\sqrt{\tan x}}$. Can you help me! , I don't have an idea for solve this problems. I think, set $t=\tan x$ but i can't solve it.
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1answer
48 views

Hint on evaluating this integral

I can't seem to come up with the proper integration technique to evaluate $$ \int_0^1 t \cdot \sqrt{\frac{2}{2-t}} dt. $$ I'd appreciate a push in the right direction!
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0answers
42 views

Is the path integral the most general representation of the inverse of the Gradient operator?

Is the path integral the most general representation of the inverse of the Gradient operator? \begin{align} \boldsymbol{\nabla} \int_{\boldsymbol{x_0}}^{\boldsymbol{x}} \boldsymbol{F} \cdot ...
9
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1answer
110 views

For which integers $1\leq{m}\leq{10}$ is it true that $\int_0^\pi{(\cos{x})(\cos{2x})\cdots(\cos{mx})}\,dx = 0$?

I could only solve this problem via brute force, trying every value from $m = 1$ to $10$... What is the more efficient and proper method of approach? (Note: my method involved repeated usage of ...
5
votes
4answers
106 views

Show that $f(x) = 1$ for all the interval

Let $f(x)$ a continuous function over $[a,b]$ Suppose that $$\frac{1}{b-a} \int_a^b (f(x))^2 \, dx = 1$$ And $$\frac{1}{b-a} \int_a^b f(x) \, dx = 1$$ Show that $f(x) = 1$ for all $x \in [a,b]$. ...
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1answer
72 views

Example for non-rectifiable set

The definition of rectifiable set is that if the constant function $1$ is integrable over the set $S$, then the set $S$ is rectifiable. What can be an example for non-rectifiable sets? I cannot ...
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0answers
22 views

Detail in Definition of Simple Functions

The motivation to this question comes from here, but it's not necessary to see the link to understand my question. In the book of Measure Theory and integration of Folland, we have the following ...
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2answers
55 views

Does there exist a green's function that does not have translation symmetry?

I noticed that most Green's functions I have used take on the following functional form $G(x_1,x_2)=G(|x_1-x_2|)$. I assume these subsets of Green's functions are translationally invariant? Correct me ...
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1answer
43 views

Order of integration.

$$ \int^1_{\frac{1}{4}} dx \int^{\frac{1}{\sqrt{x}}}_1 \frac{\sin \frac{1}{y}}{4-y^2} dy$$ I do not know how can you change the order of integration. Please help. Thanks in advance.
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0answers
48 views

Analytical Solution to an indefinite integral

I was reading about the Risch Algorithm on Wikipedia, and came across the example below, which was taken from Bronstein's "Symbolic Integration Tutorial". I do not currently have access to this ...
5
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1answer
59 views

Measure Theory - Problem with definition about simple functions

I did this question lately and then realized what my mistake was. I got a good help! But looks like only now I understand what was the real problem, to begin with. And this brought me to the same ...
1
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4answers
57 views

How to integrate $\int \frac{dx}{\sqrt{ax^2-b}}$

So my problem is to integrate $$\int \frac{dx}{\sqrt{ax^2-b}},$$ where $a,b$ are positive constants. What rule should I use here? Should substitution be used or trigonometric integrals? The ...
2
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2answers
50 views

yet another simple Laplace transform

what is $ℒ(t^2e^{3t})$ I have got this far so far: $=\int_{0}^\infty (t^2e^{t(3-s)})$ Integration by parts using: $u = t^2$ and $du = 2t$ $v = \frac{e^{t(3-2)}}{3-s}$ and $dv = e^{t(3-s)}$ Which ...
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1answer
97 views

Is this piecewise function Riemann Integrable?

Is $f(x)$ where, $ f(x) = \left\{ \begin{array}{lr} \frac{1}{x^2} & : x <0\\ x & : x \geq0 \end{array} \right. $, Riemann Integrable over $[-1,1]$? ...
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1answer
35 views

Solving this discontinuous integral using Lebesgue

Not a duplicate look at $f(x)$ here! Suppose we are to evaluate: $$I = \int_{0}^{1} f(x) dx$$ Where $$f(x)=\begin{cases}1 \space \text{if} \space x\space \text{is irrational}, & \newline 0 ...
9
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5answers
659 views

Evaluating Integrals using Lebesgue Integration

Suppose we are to evaluate: $$I = \int_{0}^{1} f(x) dx$$ Where $$f(x)=\begin{cases}1 \space \text{if} \space x\space \text{is rational}, & \newline 0 \space \text{if} \space x \space \text{is ...
0
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1answer
46 views

what is being asked here?

I fail to see how this can be achieved: Define the improper integral (of a non-blocked function) as a limit, and calculate or prove that the integral diverges; $\large \int_0^1 \frac{dx}{ ...
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3answers
132 views

Indefinite integration problem $\int {\frac{1}{1+ \tan^4 x}}dx$

The following problem came up in my last examination. $$ \int {\frac{1}{1+ \tan^4 x} dx}$$ The difficulty I was facing was that I wasn't able to find anything to substitute as there was nothing ...
1
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1answer
80 views

Separate numerator and denominator integral [duplicate]

I was given $$ \frac{\int_0^{\pi/2}\sin^{\sqrt{2}+1}(x) \, dx}{ \int_0^{\pi/2}\sin^{\sqrt{2}-1}(x) \, dx} $$ How to evaluate this integral? Since denominator and numerator are different Integral ...
5
votes
4answers
105 views

Integration: $\;\int \frac{1}{2-x^2}\,dx$

Sorry to ask about a (probably) faily easy task, but I am new to integrals and can't solve this one: I am trying to calculate $\displaystyle \int \frac{dx}{2-x^2}$. The solution is $\displaystyle ...
0
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1answer
40 views

Integration by parts with arcsin

$$\int arcsin(x)\sqrt{1-x^{2}}dx$$ I noticed that $$\sqrt{1-x^{2}}$$ is similar to the derivation of arcsinx. Is there something I can do with this piece of information? I cant just do integration by ...
4
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2answers
98 views

The value of $\lim_{n\to \infty}\int_{-\infty}^{\infty}f(x)\cos^{2} nx dx.$

Using the fact $\lim_{n\to \infty}\int_{-\infty}^{\infty}f(x)\cos nx dx=0$ ,find the value of $$\lim_{n\to \infty}\int_{-\infty}^{\infty}f(x)\cos^{2} nx dx.$$ I tried through integrating by parts , ...
4
votes
1answer
47 views

$f,g\in L^1(\mu)\implies fg\in L^1(\mu)$

Let $(X,\mu)$ be a measure space and suppose that $f,g\in L^1(\mu)$, i.e. $$\|f\|_1=\int_X|f|d\mu<\infty\quad\text{and}\quad\|g\|_1=\int_X|g|d\mu<\infty.$$ How to show that $fg\in L^1(\mu)$? ...
0
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2answers
133 views

Electric field of semi-sphere

I have to find the electrical field in the center (of the base) of a semi spherical shell of radius R. The total charge Q (Q > 0) is uniform on the intern surface of the semi sphere. Here's a scheme: ...
1
vote
1answer
69 views

Understanding the arc length integral formula

I believe the proof in my book is slighty more informal than the proof that uses the Mean Value Theorem. Could someone tell me what exactly the difference is, and if there are any mistakes in the ...
1
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1answer
35 views

Why is this 2 here (I believe I have shown it without it) - integration (Riemann integral)

I am looking at Proposition 1.3 (on page 3, how embarrassing!) The line I dispute is $I_\mathcal{P}(f)\le I_{\mathcal{P}_1}(f)+\frac{2M}{k}l(I)$ I see no need at all for the 2! Logic: ...
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5answers
130 views

$\lim_{n \to \infty}n^2\int_{1}^{\infty} \frac{cos(x/n)-1}{x^4}dx$

Show that the following limit exists and compute it: $$\lim_{n \to \infty}n^2\int_{1}^{\infty} \frac{\cos\left(\frac{x}{n}\right)-1}{x^4}\,dx$$ Attempt: By using the integration by parts, I get ...
2
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1answer
128 views

how to integrate $\displaystyle \int_{0}^{\pi/2}\frac{ dx}{1+\cos(\theta)\cos(x)}$

$\displaystyle \int_{0}^{\pi/2} \frac{ dx}{1+\cos(\theta)\cos(x)}$ where theta is in $]-\pi,\pi[$ this took me hours and i could not do it I tried using $x= \pi/2 - t$ , using the fact that ...
2
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1answer
84 views

Elliptic integral evaluation

How to integrate ( $ r_o, r_b$ constants) $$ \int \sqrt{\dfrac{r_o^2- r^2}{r^2-r_b^2}} \, dr, (r_o > r > r_b > 0)\, ? $$ With Mathematica got its coefficient imaginary, needing to take ...
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1answer
44 views

Evaluating an integral by changing the order of integration.

Problem: Evaluate $$ \int_{1/4}^1 \int_{\sqrt{x-x^2}}^{\sqrt{x}}\frac{x^2-y^2}{x^2}\, dy\,dx $$ by changing the order of integration. I have divided the region into the following three segments: ...
2
votes
3answers
188 views

Help solving an improper integral

I need to solve an improper integral which is: $$ \int_1^\infty \frac{2}{4{x^2}-1}dx $$ i was trying to solve it using simple substitution but cannot seem to figure it out, i tried a website to ...
1
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1answer
59 views

Confusion about integration notation

This is probably a silly question but I've never seen this notation: For a > 0, compute $$\int\int_{x/y \leq a} 2e^{-(2x+y)} dx dy$$ What is $x/y \leq a$ there for? This is from my statistics ...
3
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4answers
159 views

Evaluate $\int(x\sqrt{1-x^4})dx$

I've attempted this question with the substitutions $x=\sin(\theta)$ and $u = \sin^2(\theta)$ but then I got stuck. I think the main problem here is the power is too high. I'm not sure how to reduce ...
0
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1answer
53 views

why does the integral of convolution equal to the product of their integral separately?

$(f*g)(x)$ is called convolution and is the integral of $f(x-y)g(y)$ with respect to $y$ on $\mathbb{R}^n$. But why the integral of $f*g$ is equal to product of integral of $f$ and $g$. Wiki says it ...
2
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2answers
80 views

How do I evaluate $\int u^m (1-u^2)^n du$?

What I've tried so far : $$\int u^m (1-u^2)^n du$$ $$u=\sin x \implies du= \cos x dx$$ $$\int \sin^{m}x \cos^{n+1}x dx$$ I have no clue on how to continue from here. Also, if the indefinite ...
0
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1answer
43 views

Vector Integral

Let ${\bf F} = \langle y, x+2y\rangle$. Calculate $\int_C {\bf F}\cdot {\rm d}{\bf r},$ where $C$ is the upper semicircle that starts at $(0,1)$ and ends at $(2,1)$. In order to calculate this ...
3
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2answers
261 views

integral $\int \sqrt{x^4+x^3} \, dx$

$$ \int \sqrt{x^4+x^3} \, dx? $$ Using the binomial method and by setting $\frac{1}{x}+1=t$, I get to solve $$ \int \frac{-t^{\frac{3}{2}}}{(t^2-1)^4} dt? $$ It is on degree $\frac 32$. How to ...
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2answers
58 views

Natural log in indefinite integral

If we have indefinite integral and it is in the form $\frac{1}{?}$ where d(?), can we always say that the solution is $ln(?)+C$ and if not in which cases we cannot apply it?
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1answer
40 views

How to calculate integral involving Bessel function?

$$\int_0^{\infty}y^{1+\frac{m+c}{2}}K_{c-m}(2b\sqrt{y})dy,$$ How to calculate this integral? Note that, $K_{c-m}$ is the modified Bessel function of the second kind.
2
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2answers
72 views

Take integral of $\int 1/x \,dz$ where $z=x+y$?

I want to integrate $$\int \frac{1}{x} \,dz$$ where the measure is the total differential of $z=x+y$ and $x,y\in\mathbb{R}$ are variables. I wonder if the result should simply be: $$\int ...
4
votes
1answer
62 views

Evaluate a limit of series using Riemann integral

Let $$ \lim_{n\to\infty} n\cdot \sum_{j=1}^n \frac{\cos\left(\frac{n}{j}\right)f\left(\frac{n}{j}\right)}{j^2} $$ Where $f$ is $C^\infty$ and monotonically decreasing: $\lim_{x\to\infty} f(x) = 0$. ...
1
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1answer
49 views

Cannot find link between trigonometric statements and reduced form

I have been trying to find a way to reduce following trigonometric statements to the reduced form below, but without succes. I haven't been able to grasp the typical train of thought I presume I would ...
2
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1answer
91 views

finding a harmonic sum using residues/complex analysis

Evaluate: $$S = \sum_{n=1}^{\infty} \frac{H_n}{n^2}$$ Using complex analysis. I just needs hints, I have no attempts, but I believe is has to do with residues.
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1answer
83 views

Help for Integral and evaluating - Eikonal equation

Hy guys I'm reading a paper of "Finding Exact Solutions to the Two- Dimensional Eikonal Equation" - E.D. Moskalensky. link for the paper: ...
0
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1answer
49 views

Weird vector projection form

Let $C^0[1,3]$ be the $\Bbb R$-vector space equipped with the usual scalar product (by the integral ). Calculate the projection of the function $f(x)= 1/x$ onto the subspace $W = L\{x\}$ Well, my ...
1
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1answer
28 views

Volume of $ x^2-3 \le y \le 1 $ around $ y = 2 $

I need to calculate the volume of $$ x^2-3 \le y \le 1 $$ around $$ y = 2 $$ My initial thought was to try $$ {2 \cdot \pi} \int_0^2 (x^2-3)^2 \,dx $$ But then it hit me that because the ...
0
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3answers
49 views

Max or min of $F(x) = \int_0^{2x-x^2} \cos\Big(\frac {1}{1+t^2}\Big) \,dt$

$$F(x) = \int_0^{2x-x^2} \cos\left(\frac {1}{1+t^2}\right) \,dt$$ Does the function have a max or min? Can someone help me with this? How can I calculate the maximum and minimum?
1
vote
2answers
46 views

How should terms be scaled by finite dx and dt in numerical integration of 1D diffusion?

I am familiar with numerically integrating systems of ordinary-differential equations, but I feel that I am missing something important in terms of how numerically integrating ODEs differs from ...