Questions tagged [integration]

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 the integrals being considered. This tag often goes along with the (calculus) tag.

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0answers
23 views

Calculate $\int_{0}^{1} \int_{y}^{1} e^{-x^2} \, \mathrm{d} x \mathrm{d}y$

Calculate $\displaystyle \int_{0}^{1} \int_{y}^{1} e^{-x^2} \, \mathrm{d} x \mathrm{d}y $ I understand that multiple integral has to be used, but I can't go further. Can anyone show me how to solve ...
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1answer
22 views

Find the Area under $R$ Using Integrals

$R$ is a region in the first octant of $\mathbb{R^3}$ that is bounded by $x^2+y^2+z^2=4$ and $z=\sqrt{x^2+y^2}$. Set up the integral for the volume of this region $R$. I think the first thing to do is ...
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0answers
15 views

Solution to $\int\frac{1}{x+xe^{ax+b}}dx$

I need to solve $\int\frac{1}{x+xe^{ax+b}}dx$ as part of a larger optimization problem, where the $a$ and $b$ parameters will be optimized in a subsequent step. Wolfram alpha returns the following ...
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0answers
6 views

U-substitution in proof of Frullani's Theorem

In a proof of Frullani's Theorem I'm trying to understand there's a step where $$\int_x^y \frac{f(at)}{t}dt - \int_x^y \frac{f(bt)}{t}dt = \int_{ax}^{ay} \frac{f(u)}{u/a} \frac{du}{a} - \int_{bx}^{by} ...
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1answer
31 views

Not sure how to finish this integral $\int_{-\infty}^\infty x^3 \delta(x^2-2)dx$

Dirac delta is a symmetric function defined as $$\int_{-\infty}^\infty f(t)\delta(t-A)dt = f(A)$$ Find the value of $$\int_{-\infty}^\infty x^3 \delta(x^2-2)dx$$ SOLUTION: Let $t=x^2$ then $dt=2xdx \...
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0answers
30 views

Is it possible to get a closed form expression for $E\left(\dfrac{1}{a+e^Z}\right)$ for $Z\sim N(0,1)$ and $a>0$?

The question is in the title. So all we need to do is compute the integral $$\int_{-\infty}^\infty \dfrac{e^{-x^2/2}}{a+e^x}dx$$ I tried some integration by parts tricks but could not proceed. To be ...
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38 views

Proof that $\int f(x-a)\cdot g(x)dx = \int f(x)\cdot g(x+a) dx$

How can one proof that: \begin{equation} \int f(x-a)\cdot g(x)dx = \int f(x)\cdot g(x+a) dx \end{equation} and on what boundaries would this hold (I assume only $[-\infty, \infty]$)?
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22 views

Different integral notations

In MS Word there are 2 different integral notations as shown in the screenshot below: Is there any difference or it is the same?
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33 views

Solving for real values of a quadriatic equation

I have already fit a quadratic curve $ax^2 + bx + c$ to experimental data and calculated the value of its coefficients (a, b, c). Now, I have to compute two integral functions (#14 and #17) as shown ...
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23 views

All the functions that satisfy an integral condition

I need to find a caractherize the function $A(a, b)$ such that \begin{align} \int_0^xA(x+y, z-\frac{x+y}{2})z dz+\int_0^yA(x+y, z-\frac{x+y}{2})z dz=\alpha x^2+\alpha y^2+\beta xy, \end{align} where $\...
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28 views

How to solve this integral depending on parameter

$g(\alpha )=\int_{a}^{b} \frac{f(x)}{\sqrt{\left | x-\alpha \right | } } dx$ Given that f(x) is bounded, the question asked me to prove g(α) is continuous on R. The question is, I don't even know if ...
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22 views

integration of product of Exponential and square of Bessel function and x

I need solution of above integral
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24 views

Representation of a mollified Coulomb kernel through integration over a sphere

I am looking for a result that allows represent a function through the integral over a sphere. So, does anybody knows a proof of the following statement? Given $R>0$ we define for every $x\in \...
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0answers
35 views

Why can't I use Cauchy-Schwarz to show convergence of the Fourier series in $L^1$?

Recently I read that there are $L^1(\mathbb{T})$ functions such that their Fourier series does not converge in the norm of $L^1$, that is if $S_N(f)(x)$ represents the $N$-nth partial sum of the ...
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0answers
46 views

Double integral $\frac{1}{\sqrt{x^2+y^2}} e^{i(ax+by)}$

How can I integrate $$\int \int \frac{1}{\sqrt{x^2+y^2}} e^{i(ax+by)} dx dy$$ over $\mathbb{R}^2$, where $a, b$ are constants?
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50 views

Integration of differential form inclusion of $S^2$

Take this example. We have the natural inclusion $i : S^2 \rightarrow R^3$ and the differential form: $\omega = x dy \wedge dx + y dz \wedge dx + z dx \wedge dy $. How can we say that that the ...
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32 views

Why is this curve the integral curve of the described vector field? [closed]

In Lee's booka bout manifolds we have this example: I am trying to udnerstand how we can so easily jump from $x^2\frac{\partial}{\partial x}$ to $\gamma(t)$.
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1answer
68 views

Calculate $\int_{C} e^{\frac{z^2}{z^2-1}}dz$

Calculate $$\int_{C} e^{\frac{z^2}{z^2-1}}dz$$ where $C$ is the circle of radius $3$ centered at $0$ oriented counter clockwise. So I want to do this by finding $\operatorname{Res}(f,\infty)=-\...
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2answers
127 views

How can i evaluate this integral with or without using CAS $\int_0^{\infty}\frac{\mathrm dx}{(x^2+1)\cosh(\pi x)}$

$$\int_0^{\infty}\frac{\mathrm dx}{(x^2+1)\cosh(\pi x)}$$ Firstly i used substitution $\pi x=t; \mathrm dx=\frac{\mathrm dt}{\pi}$ $$\pi \int_0^{\infty}\frac{\mathrm dt}{(t^2+\pi^2)\cosh t}$$ writing $...
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1answer
29 views

Prove that a bounded $f$ is integrable if $I_0 := \lim_{n\to\infty}L(f,P_n) = \lim_{n\to\infty}U(f,P_n)$

Prove that a bounded function $f$ is integrable on $[0,1]$ if $$I_0 := \lim_{n\to\infty}L(f,P_n) = \lim_{n\to\infty}U(f,P_n),$$ in which case $\int_0^1f(x)dx$ equals $I_0$. Refer here. I suspect that ...
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0answers
37 views

Solve $\int _{|z-1|=1} \frac{\log{z}}{z^{4}} dz $

this problem came in my exam today and I couldn't solve it. This is my failed attempt, using Cauchy's Integral Formula for derivatives. I think that the theorem tells me that if $z_ {0} \in $ my curve ...
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2answers
40 views

Is $\lim_{t\to\infty} e^{-at} \int_0^t e^{a t'} f(t') dt'$ ($a>0$) zero?

Given that $f(t)$ is a well-behaved positive function, $f(0)=1$ and $f(t)$ monotonically decreases towards zero as $t\to\infty$. Can one derive the limit of the expression? I "feel" the ...
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1answer
51 views

Simplifying the integral $\int \frac{x^2-1}{x^2+1}\cdot \frac{1}{\sqrt{1+x^4}}dx$.

I have the following integral that I must simplify by making two substitutions \begin{align} \int \frac{x^2-1}{x^2+1}\cdot \frac{1}{\sqrt{1+x^4}}dx, \end{align} one of them is \begin{align} y = \frac{...
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2answers
36 views

Function $f$ with $|f|$ is Lebesgue integrable but $f$ isn't locally Lebesgue integrable

I'm trying to find a function $f$ with $|f|$ is Lebesgue integrable but $f$ isn't locally Lebesgue integrable. My approach: Let $X = [0,1]$ and $A \subset X$ with A is the Vitali-set. So $A$ is not ...
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4answers
92 views

Asymptotic behaviour of $\int_{0}^{1}f(x)x^ndx $ as $n\to \infty $

I'd like to determine the asymptotic behaviour of $$\int_0^1 f(x)x^n\,dx $$ as A) $n\to \infty $ with $0<f(x)\in \mathrm{C}^1([0,1]), 0\le x\le 1$ B) $n\to \infty $ with $0<f(x)\in \mathrm{C}^0([...
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0answers
18 views

Γ(z)=M{e^-x}(z) How to write it expanded form? Whether M{e^-x} is constant or complex number when z is complex number. [closed]

Γ(z)=M{e^-x}(z) How to write it expanded form? Whether M{e^-x} is constant or complex number when z is complex number. Γ(z)= ( z-1)! = a complex number result arrived always. Example (z -1) ! =. ( a +...
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0answers
21 views

How to minimize some integral by choosing an optimal density function?

The problem I would like to solve is as follows: $$ \min_{\substack{\hat{f} \\ \hat{f} \geq 0\\ \int\hat{f} \;dw = 1}} \int_{\mathcal{Z}} \left( \int_{\mathcal{W}}\left[\hat{g}(z|w)\hat{f}(w) - g(z|w)...
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0answers
31 views

Help with Integral Involving fractions and exponentials

Hi I need help with the integral below. I have had a tried a few substitutions but they don't work. $\int_{z_u}^{z_f} \frac{dz}{\sqrt{\frac{5+\gamma e^{-bz}}{\delta^2}-6}}$ where $\delta$, $\gamma$ $a$...
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0answers
72 views

Is there an antiderivative for $e^{-\left( x + \frac{1}{x}\right)}$?

I was playing around with some integrals I made up myself and was trying to find a closed-form for $$ \int_{0}^{t} e^{-\left( x + \frac{1}{x}\right)} \ dx, \qquad t < \infty $$ I'm aware that if ...
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1answer
47 views

for what values of parameters p,q does the integral $\int_1^{\infty} \frac{1}{x^p + x^q}dx$ converges?

What I thought is to break this into different cases: $p=0$: so the integral given ~ $\displaystyle \int_1^{\infty}\frac{dx}{x^q}$ , and it converges for $q>1$. $p<0$: so the integral given ~ $\...
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1answer
42 views

Summation by Part and Integration by part.

How is summation by part similar to integration by parts? How are they different? I was doing some questions on integration by parts and summation by parts and now i am curious to know the ...
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0answers
37 views

why is the geometric interpretation for integral $\int_c^d \frac{1}{\sqrt{(x-a)(x-b)}}$ a degree?

why is the geometric interpretation for integral $\int_c^d \frac{1}{\sqrt{(x-a)(x-b)}}$ a degree? I once saw that an intuitive geometric interpretation is as follows: This integral reflects the ...
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0answers
32 views

Average value of $\sqrt{s^2-x^2-y^2}$

This is a function $f(x,y)=\sqrt{s^2-x^2-y^2}$. I want to show that the average value of this function is $\dfrac{2}{3}s$ on the region $D=\{(x,y):x^2+y^2\leq s^2\}$, $s \geq 0$. Assuming that $f(x,y)$...
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2answers
63 views

Using Leibniz's Rule to Evaluate Integrals

Problem Evaluate the integral $$\int_0^1 \frac{\ln(x+1)}{x^2+1} \, dx$$ using Leibniz's rule. Solution attempt. To evaluate the above integral, we consider a similar integral $$F(y) = \int_0^1 \frac{\...
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0answers
34 views
+100

Bounding the propagation function

For $\alpha \in (0,1)$, set $\omega:\mathbb{R^+}\times \mathbb{R^+} \to \mathbb{R}$ defined as following $$\omega(t;\tau):=1-\pi^{-1}\int_0^\infty \frac{e^{-rt-\tau r^\alpha \cos(\alpha \pi)}\sin(\tau ...
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1answer
92 views

Techniques for approximating $\int_{-\pi}^\pi \left(\frac{\sin x}{x}\right)^{30} dx$ [closed]

What are the techniques to approximate the value of this integral? $$\int_{-\pi}^\pi \left(\frac{\sin x}{x}\right)^{30} dx$$ I have no clue how to go about it.
1
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1answer
82 views

Theorem 6.12(a) Of Baby Rudin. Alternative Proof Of $ \int_a^b \left( f_1 + f_2 \right) d \alpha = \int_a^b f_1 d \alpha + \int_a^b f_2 d \alpha$

If $f_1 \in \mathscr{R}(\alpha)$ and $f_2 \in \mathscr{R}(\alpha)$, then $$ \int_a^b \left( f_1 + f_2 \right) d \alpha = \int_a^b f_1 d \alpha + \int_a^b f_2 d \alpha$$ I want to prove this theorem ...
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0answers
26 views

Is there a proof for this Theorem related to distributions and almost everywhere equality to 0?

I came across this theorem in one of my courses on Sobolev spaces. I am struggling to find a clear proof for it and I have not succeeded yet. Some notation first: $L_{loc}^{1}\left( \Omega \right) $ ...
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1answer
49 views

Definite Integration With Trig Functions and Euler's Number

I'm trying to calculate a welfare function as a definite integral, and the part I'm having trouble with is the consumer curve function. In econ, this is done as $\text{CS}=\int_{0}^{Q^{\ast}} \left[f \...
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0answers
48 views

Show Riemann-integrability of $h:=\max\{f,g\}$

Let be $f,g$ two Riemann-integrable functions on the interval $[a,b]$ where $a<b$. Show that $h:=\max\{f,g\}$ is also Riemann-integrable on $[a,b]$. My approach: First, a bit of notation: \begin{...
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1answer
47 views

Do we still add $C$ when integrating and the result is an arbitrary function?

I apologize if this question sounds stupid. I'm just bothered whether I should write $$u(x,y) = f(x) + C$$ or just $$u(x,y) = f(x)$$ For the solution of this problem: $$\frac{\partial u(x,y)}{\partial ...
1
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1answer
129 views

Convergence of $\int_0^\infty\frac{\sin x}{x}dx$

I'm working on proving the convergence of the below integral, the value of which I know to be $\displaystyle \frac{\pi}{2}$. $$ \int_0^\infty\frac{\sin x}{x}dx $$ I'm also well aware that this ...
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0answers
37 views

First semester in University Integral Inequality Question [closed]

I have problems with proving the following inequality, which is the last exercise of the chapter of our University book regarding measuring length, area and volume with integrals. $$ \sqrt{(b-a)^2+(f(...
5
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1answer
53 views

Theorem 10.2 Rudin

$\mathscr b(X)$ denotes the set of all complex-valued, continuous, bounded functions with domain $X$. I don't understand why is $L(h)$ equal of $\prod_{i=1}^k $ $\int_{a_i}^{b_i} h_i(x_i)dx_i$ and ...
2
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0answers
35 views

Baby Rudin 10.1

I want to be sure that I understand it correctly. As I understand it $\int_{I^k} f(x) dx $ = $\int_{a_{k-1}}^{b_{k-1}}$ ($\int_{a_k}^{b_k} f_k(x_1,...x_{k-1},x_k)dx_k$)$dx_{k-1}$ and to continue this ...
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0answers
50 views

Evaluate $I=\int_{0}^{1}\operatorname{Li}_2^2(x) \operatorname{Li}_3^2(-x)\ln(1+x)\text{d}x$

Denote $$ \omega_0(t)=\frac{\text{d}t}{t} , \omega_1(t)=\frac{\text{d}t}{1-t}, \omega_2(t)=\frac{\text{d}t}{1+t}. $$ I want to compute($\text{Li}$ denotes polylogarithms) $$I=\int_{0}^{1}\operatorname{...
2
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0answers
60 views

Help proving that signed volume of n-parallelepiped is multilinear

Overview I am trying to build some intuition about the volumes of parallelepipeds and determinants. I would like to define the determinant as the unique function of $N$ vectors in $\mathbb{R}^N$ which ...
0
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1answer
69 views

Evaluate $\iint_D xy$ [closed]

My question is, evaluate the area under the curve, $\iint_D xy$, where $D$ is a region bounded by $y=\sin(x)$ and $y=\cos(x)$. Here is an illustration of it: As you can see, $D$ is the area between ...
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1answer
115 views

Where I made mistake(s) as the integrating of $~\int_{}^{}\frac{1}{\sin^{2}\left(x\right)}\,dx~$?

$$L:=\int_{}^{}\frac{1}{\sin^{2}\left(x\right)}\,dx\tag{1}$$ $$=\int_{}^{}\sin^{-2}\left(x\right)\,dx\tag{2}$$ $$=\frac{\sin\left(x\right)^{-1}}{\left(-1\right)\left(\cos^{}\left(x\right)\right)}+\...
0
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1answer
29 views

I need the expression of a function which goes to infinity as x goes to 0 and has a finite integral

Is there a function similar to the one in the image that has a discontinuity in the origin, for which: $$\lim_{x \to \pm\infty} f(x)=0$$ and $$\lim_{x \to 0^{\pm}} f(x)=+\infty$$ and $$\int_{-\infty}^{...

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