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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 ...

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8 views

Prove that this integral with exponent with whole part is convergent

How can i prove that this integral is convergent? $$ \int_{\vert\xi\vert\geq 1} \vert\xi\vert ^{-2\left[ \frac{n}{2}\right] -2} d\xi $$ I thought to use polar coordinates but i don't know how to ...
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14 views

Simpson vs. trapezoidal rule for numerically integrating cos(x)*cosh(x) in range 0 to pi?

I have to numerically calculate many integrals similar to this: $$\int_0^\pi \cosh{\left(\frac{a_1\cos{x}+a_2\cos{2x}+a_3\cos{3x}+\ldots}{10}\right)}\cos{jx}\cos{kx}\space dx$$ Right now I am using ...
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How to integrate$\int_0^{t} xe^{i(ax^2)}J_0(bx)\,\mathrm dx$?

Find $$\int_0^{t} xe^{i(ax^2)}J_0(bx)\,\mathrm dx$$ Given $a>0,\ b>0, t>0$. Previously I asked a more difficult integration problem. It seems that it is too difficult to find a solution. ...
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2answers
29 views

Integration by parts yields divergent integral

$$I = \int_0^1\dfrac{t \ln t}{\sqrt{1-t^2}}\mathrm{dt}$$ Attempt: Let $t = \sin x$ $\implies dt = \cos x dx$ $\implies I = \displaystyle \int_0^{\pi/2} \dfrac{\sin x \ln (\sin x) \cos x dx}{\cos ...
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18 views

Points of Confusion About Second-Order Taylor Formula of Taylor's Theorem For Many Variables

My textbook has written the following for the second-order Taylor formula of Taylor's theorem for many variables: $$f(\mathbf{x}_0 + \mathbf{h}) = f(\mathbf{x_0}) + \sum_{i = 1}^n h_i \dfrac{\...
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1answer
16 views

why does the variable bound (i.e. the bound that changes) must always be upper bound

I'm currently learning integral calculus, in particularly, integral function with the lower bound been the variable and upper bound is the constant. Below is a question and solution from khan ...
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1answer
48 views

Integral of $\sin^5(x)\cos(x)$

I'm trying to solve the following integral: $$\int\sin^5(x)\cos(x)$$ I assumed I would do u-substitution where: $$u = \sin(x)$$ $$du = \cos(x) dx$$ Which would then cancel out the $\cos(x)$ And ...
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13 views

Regarding moving the differential inside of a double integral.

Forgive me it has been a few years since I took multivariable calculus so I am unsure about something. Say I am given something like $\int_x^y \int_x^y h(a,b)f(a,b|x,y)dadb = g(x,y)$. Specific ...
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Vector integration to find electric Field

How can i solve this integral to find electrical field E using spherical coordinate system?[1]: https://i.stack.imgur.com/yn8h3.png
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1answer
16 views

Rotating a figure around x-axis whose area is given

Let's say I have a 2d-figure above the x-axis and I'm already given it's area (so I don't have to integrate). All of this figure touches the x-axis, so there's no hole in the middle when rotating How ...
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25 views

How to go about this (double integral?) problem?

This question concerns the derivation of equations for calculating (1) the flow into a conically-shaped wellbore and (2) the inner cylindrical radius that would provide flow into a cylindrically-...
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1answer
40 views

Polar integral $\int_{r=0}^{\infty} \frac{b r dr}{(r^2+b^2)^{(3/2)}(r^2+a^2)^{(1/2)}}$

We need to solve above integral which I obtained after manipulation. Now I used $r^2 = u $ to get: $$\int_0^\infty \frac{bdu}{2 (u+b^2)^{(3/2)}(u+a^2)^{(1/2)}}$$ which is converted to: $$\int_0^\...
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1answer
37 views

Evaluate $\oint_L (x-y)dx+xydy$

Evaluate $$\oint_L (x-y)dx+xydy$$ where $ L\in \{x=a\cdot \cos ^3t,y=a \cdot \sin ^3t\}, 0\leq t \leq \Pi$ and $\{x=a\cdot \cos t,y=a\cdot \sin t\}, \Pi\leq t \leq 2\Pi $ So I try $$=\int_0^\Pi ((a\...
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2answers
51 views

Proof inequality $\frac{\sqrt{\pi}}{2}\sqrt{1-e^{-a^2}} < \int_0^a e^{-x^2}dx < \frac{\sqrt{\pi}}{2}\sqrt{1-e^{-2a^2}}$

I'm asked to prove the inequality $$\frac{\sqrt{\pi}}{2}\sqrt{1-e^{-a^2}} < \int_0^a e^{-x^2}dx < \frac{\sqrt{\pi}}{2}\sqrt{1-e^{-2a^2}}$$ After playing around for a while I was able to find (...
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32 views

Non-infinitesimal differential operators, do they exist?

I already know they do exist in signal processing, I have built many myself. It is a whole art to do so in any discipline that handles noisy data. But is it possible to find out which ...
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0answers
29 views

How to integrate $\int_0^{t} xe^{i(a x^4-bx^2)}J_0(cx)\,dx$?

Find $$\int_0^{t} xe^{i(a x^4-bx^2)}J_0(cx)\,\mathrm dx$$ Given $a>0,\ b>0, c>0, t>0$. I decompose it into two parts: $$\int_0^{t} x\cos{(a x^4-bx^2)}J_0(cx)\,\mathrm dx,$$ $$\int_0^{t}...
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2answers
55 views

Integrate $ \int \frac{x^4 +1}{x^6 - 1}dx $

Integrate $$ \int \frac{x^4 +1}{x^6 - 1}\, \mathrm dx$$ I have tried using partial fractions but to no use. Thanks for help.
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1answer
27 views

Properties of Riemann integral $\left| \int_b^{a}f\right| \lt \int_b^{a}|f|$?

$$f(x) \lt |f(x)|\implies\int_b^{a}f \lt \int_b^{a}|f|,$$ $$-f(x) \lt |f(x)|\implies-\int_b^{a}f \lt \int_b^{a}|f|.$$ Hence , $$-\int_b^{a}|f| \lt \int_b^{a}f \lt \int_b^{a}|f|.$$ Therefore $$\...
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1answer
20 views

Given $f:[0,+\infty]\to\Bbb [1,+\infty)$, if $1/f\in\mathcal{R}[0,+\infty)$ then is $f$ not Lipschtiz?

I thought of proceeding by contrapositive. Assume $f(x)\ge1$ and $f'(x)\le K\in\Bbb R$ for all $x\ge0$. Then for any $t>0$ it follows that \begin{align} \int_0^tf'(x)\ dx=f(t)-f(0)\le Kt\end{align}...
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30 views

Solve Surface Integral

I have been given this question to solve however I'm having some difficulty solving it as I am quite new to Partial Differential Equations: Let $ a = (2,2) $ and $ r =5 $ Compute the following ...
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58 views

Integration by parts for definite integral

$$\int_{0}^{\frac{\pi}{2}} \frac{x^2}{(x\sin\ x+\cos\ x)^2}=\left[\frac{-x\sec\ x}{(x\sin\ x+\cos x)^2}\right]_0^\frac{\pi}{2}+\int_{0}^{\frac{\pi}{2}} \sec^2 x dx$$ Is this a correct application of ...
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19 views

Area of one loop of the curve $r=5\cos4\theta$

Area of one loop of the curve $r=5\cos4\theta$ The area of one loop can be given by: $\frac{1}{2}\int^\frac{\pi}{8}_\frac{-\pi}{8}(5\cos4\theta)^2\,d\theta$. However this gives me an answer of $\frac{...
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1answer
36 views

Taylor's Theorem Proof: Confused About the Validity of Two Inequalities

My textbook gives the following proof of the single-variable version of Taylor's theorem: As promised, we begin with the Fundamental Theorem of Calculus, written in the form $$f(x_0 + h) = f(...
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1answer
24 views

Continuous Transformation of $u$-Substitution

$u$-substitution tells us that $$ \int_{0}^{\pi/2}(\sin x)^2\cos x\,dx = \int_{0}^{1}u^2\,du = \frac{1}{3}, $$ where we made the substitution $u(x) = \sin x$, and $u(0)=0$, $u(\pi/2)=1$. What I ...
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1answer
58 views

$\int_{-\infty}^{\infty}\frac{\arctan\sqrt{x^2+2}}{(x^2+1)\sqrt{x^2+2}}dx=\zeta(2)$ Without Feynman Integration

How do I find $$\int_{-\infty}^{\infty}\frac{\arctan\sqrt{x^2+2}}{(x^2+1)\sqrt{x^2+2}}dx=\zeta(2)$$ without Feynman integration? I saw this video, which gives $$\int_{0}^{1}\frac{\arctan\sqrt{x^2+2}}{(...
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0answers
33 views

Limit of integral involving Dirac Delta

I have to compute the following integral $$\lim_{\epsilon \rightarrow 0} \int_a^\infty \delta_\epsilon (x) f(x) dx$$ where $\delta_\epsilon$(x) is such that $\int_{-\infty}^\infty \delta_\epsilon (x)...
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1answer
40 views

Does an example of a function satisfying these properties exist?

Do there exist functions $f: \mathbb{R} \to \mathbb{R}$ with the following properties? 1: $f$ is at least $C^3$ differentiable. 2: $f$ and at least its first three derivatives vanish at infinity. 3:...
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1answer
40 views

Integral Expression Without $dx$, $dy$, $d\tau$, etc.

My textbook gives the following proof of the single-variable version of Taylor's theorem: As promised, we begin with the Fundamental Theorem of Calculus, written in the form $$f(x_0 + h) = f(...
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0answers
43 views

Calculating integral of absolute integrand

I want to calculate the integral $$ \int_0^1\int_0^1\int_0^1 {\rm d}r_1{\rm d}r_2{\rm d}r_3 \, r_1 r_2 r_3\int_0^\pi \int_0^\pi \int_0^\pi {\rm d}\phi_1{\rm d}\phi_2{\rm d}\phi_3 \\ \left| r_1r_2\sin(\...
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1answer
31 views

$\iint (x+y)e^{xy} dA$ . where A is region enclosed by $x-y = 1$, $x-y=4$, $xy = 1$, $xy=2$

Find the integral $\iint (x+y)e^{xy} dA$ . where A is region enclosed by $x-y = 1$, $x-y=4$, $xy = 1$, $xy=2$ So I tried the following substitution: $$u = xy$$ $$v = x-y$$ And want to find the ...
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1answer
29 views

Measure theory and almost surely

Is it correct to say that if $E[z] < \infty$ then $z$ is almost surely finite?
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1answer
24 views

Understanding how to compute the space volume for an ideal gas

There is a system of N non-interacting particles (Ideal Gas). The Hamiltonian of a system of free particles is given by: $$H = \sum_{i=1}^{N}\frac{p_{i}^2}{2m} + \sum_{i=1}^{N} \psi(q_i)$$ where to ...
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26 views

Definite integration of spherical Bessel function of radical argument

I have to solve some integrals of the form: $$\int_0^{x_0} dx \, j_n( R ) \cdot \frac{p(x)}{R^n}$$ where $R=\sqrt{x^2 + 2 a c x + c^2}$ , $j_n$ is the spherical Bessel function of order n, p(x) is a ...
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12 views

Integration by parts (proposition in Bontemps et al (2000) “Equilbrium Search with Continuous Productivity Dispersion”)

I am struggling to understand a step in a proof of existence of equilibrium in the following paper: Bontemps, Robin, Berg (2000) "Equilbrium Search with Continuous Productivity Dispersion" (starts at ...
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2answers
59 views

Calculus u-substitution: Choosing between $\frac{dx}{du}$ and $\frac{du}{dx}$

I searched and couldn't find any answers to this one. It's probably a dumb question but one that has been troubling me. Let's say I have an integration: $$\int \frac{1}{3x+2}$$ It seems the correct ...
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1answer
22 views

How to use the Divergence Theorem in this question?

Question: Let $r=\sqrt {x^2+y^2+z^2}$ and $\mathbf E = -\mathbf \nabla \big(\frac kr \big)$ where $k$ is a constant. Show that $$ \iint_S \mathbf E \cdot d \mathbf S = 4\pi k$$ where $S$ is any ...
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1answer
50 views

Calculate the surface area with integration

Calculate the surface area of the surface obtained when the region enclosed by the given curves is revolved about the $x$-axis $$y=2x^2-8$$ $$y=x^2-1$$ This is a model problem for an exam and I ...
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2answers
22 views

Ho do I got about solving this integral for a variable?

I am currently in an introductory course to quantum mechanics. For a homework problem we are given the wave function $$ \Psi(x) = \frac{N}{x^2 + a^2}, a > 0 $$ and asked to normalize it ("Solve ...
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1answer
34 views

Line integral in a vector field

I have the following question: Determine the total flux of the velocity field $$\vec{v} = \begin{pmatrix} 1 \\ x + y \end{pmatrix}$$ out of the disk $D$, a circle with radius 2 and center at the ...
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2answers
108 views

Convergence of series $\sum\int\limits_{1}^{+\infty}e^{-x^n}\,dx$

Determine if the series $\sum \alpha_n$ converge, where: $$\alpha_n=\int\limits_{1}^{+\infty}e^{-x^n}\,dx.$$ Attempt. Ι am pretty sure the inequalities $$e^{-x^n}\leq \frac{1}{1+x^n}$$ and $e^{-...
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2answers
116 views

How to integrate $\int_{-\infty}^{+\infty} \exp(-\sqrt{a^2+z^2}) dz$

$$\int_{-\infty}^{+\infty} \exp(-\sqrt{a^2+z^2}) dz$$ According to info provided by @skbmoore: Gradshteyn, Izrail Solomonovich, and Iosif Moiseevich Ryzhik. Table of integrals, series, and products. ...
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137 views

Evaluating $\int_0^1\ln(1+x^2)\ln(x^2+x^3)\frac{dx}{1+x^2}$

How to evaluate $$I=\int_0^1\ln(1+x^2)\ln(x^2+x^3)\frac{dx}{1+x^2}?$$ It equals $\frac5{64}\pi^3-\frac92G\ln2+\frac14\pi\ln^22$ according to Mathematica, where $G$ denotes Catalan's constant. Attempt ...
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1answer
51 views

Separation of Variables in Integration. Why is it necessary?

I'm quite a novice with differential equations and such and english isn't my first language so I ask to please explain everything in simple english terms. I am currently studying differentiation ...
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14 views

Under which conditions the following infinite integral involving a zeroth order Bessel function is convergent / defined?

Everybody hello, The goal is to determine the non-trivial conditions between the real parameters $a$, $b$, and $c$, for which infinite integral below is convergent / defined: $$ \int_0^\infty q J_0(...
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1answer
37 views

Intuition behind the definition of line curve

According to wikipedia If $X = \mathbb{R}^{n}$ is the $n$-dimensional Euclidean space, and if $\gamma$: $[a,b]\rightarrow\mathbb{R}^{n}$ is an injective and continuously differentiable function, then ...
2
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1answer
67 views

What is the value of the following definite integral?

$I=\displaystyle\int\limits_0^\infty \dfrac{1}{(1+x^2)\displaystyle\sqrt{\log(1+x)}}dx$ My attempt: Substituting $x=\tan{\theta}$ $I=\displaystyle\int\limits_0^{\frac{\pi}{2}} \dfrac{1}{\...
4
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1answer
120 views

Integration of $\int^{\pi/45}_0 \frac{x^2 \ln (1-x)}{1-x^3}\, dx$

How would you integrate: $$I=\int^{\pi/45}_0 \frac{x^2 \ln (1-x)}{1-x^3} dx$$ My attempt: $$du=\frac{x^2}{1-x^3}dx ⇒ u=\frac{-1}{3}\ln (1-x^3)$$ $$v= \ln (1-x) ⇒ dv=\frac{-1}{1-x}$$ $$I=\frac{-1}{...
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2answers
77 views

Why CDF is not coming to 1, when i put the maximum range?

Suppose, $F_X(x)=-\frac{x}{a^2}+\frac{2\sqrt{x}}{a}$ And, $f_X(x)=\frac{d}{dx}F_X(x)=\frac{1}{a\sqrt{x}}-\frac{1}{a^2}$, Here, $0\leq x \leq a^2$ Similar, $f_Y(y)=\frac{1}{a\sqrt{y}}-\frac{1}{a^2}$, ...
2
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0answers
14 views

Inequality to study for the solution of a linear wave equation

I'm studying a paper about a linear problem concerning with the wave equation. Starting, the author says that this inequality will be used: $$ \int_{0}^{\delta} \vert \xi\vert ^{k} e^{-c \vert\xi\...
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0answers
32 views

How to find the Volume of Dubai's Museum of the Future (An ellipsoid with an ellipsoidal hole in it)?

I'm an IB Mathematics HL student doing a math research report for school, I need to be able to model Dubai's Museum of the Future building. Do you know how I could model this shape mathematically and ...