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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1
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1answer
22 views

$\int_{\Omega\setminus A_n}f\;d\mu\to\int_\Omega f\;d\mu$ for all measurable $A_n\downarrow\emptyset$

Let $(\Omega,\mathcal{A},\mu)$ be a measure space $(A_n)_{n\in\mathbb{N}}\subseteq\mathcal{A}$ such that $A_n\downarrow\emptyset$, i.e. $A_n\supseteq A_{n+1}$ and ...
3
votes
3answers
235 views

Integral involving Bessel functions of the first kind

I am stuck with the following integral. Does it converge? $$ \int_{0}^{\infty}\left(J_1(x)^2+J_1(x)J_1(x)^{''}\right)\text{d}x $$ According to tables I find that the first term is divergent, so I ...
0
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0answers
26 views

Derivatives of semi-elasticities

I want to find the sign of $\partial_z (\partial_z \log F(T1) - \partial_z \log F(T2) )$ where $T1<T2$ and $F(T) = \int_0^T e^{g(t) + z\, h(t)} dt$ All we know is that $h(t)>0\forall t$ ...
0
votes
1answer
50 views

Integral of a total derivative

I have seen the "total differential" $$ d \ln A = -d \ln B/c $$ Representing how infinitesimal changes in $A$ are related to infinitesimal changes in $B$. Someone then took the integral of this ...
0
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0answers
9 views

Accurate numerical integration for “data times an analytical function”

The Question is as follows: I have an algorithm/data that provides me the value of a function $f(x,y,z)$ on the points of a grid. On the other hand I have an analytical function ...
1
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2answers
27 views

Evaluating integrals with trigonometric function

Now I have to evaluate the integrals $$ \int_{0}^{\pi /2} \sin^2 t \cos t dt $$ $$ \int_{0}^{\pi /2} \cos t \sin^2 t dt $$ $$ \int_{0}^{\pi /2} \tan^2 t dt $$ For the first two integrals, I could ...
2
votes
3answers
47 views

Find the value of the integral $\int_0 ^\sqrt2 \sqrt{4-x^2} \ dx$

Find the value of the integral $\int_0 ^\sqrt2 \sqrt{4-x^2} \ dx$ I was unsure how to do this question so I looked at the mark scheme, and it said use $x=2\sin\theta$ and so $dx=2\cos\theta \ ...
0
votes
1answer
31 views

Undefined Subintervals - Riemann Integrals

I searched through stackexchange and multiple other PDFs but couldn't find an answer I'm curious to know when talking about Riemann Integrals with respect to functions that are bounded on closed ...
0
votes
0answers
41 views

Help with setting up this double integral [on hold]

"By evaluating an appropriate double integral, find the volume of the wedge lying between the planes $z=px$ and $z=qx$ (with $p>q>0$) and the cylinder $x^2+y^2 =2ax$ (where $a > 0$)." I'm ...
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0answers
30 views
1
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0answers
38 views

Need some hints to solve a problem from “Revue de la filière mathématique”

Let $A\in M_{n}(\Bbb R)$ and $B\in M_{n,m}(\Bbb R)$ and $C=\int_{0}^{1}\exp\left(sA\right)BB^T\exp\left(sA^T\right)\,{\rm d}s$. Prove that $C$ is invertible if and only if $\sum_{i=0}^{n-1} ...
2
votes
4answers
72 views

How to find $\int \frac{\ln(x)}{x^2}dx$

I need to find $$\int \frac{\ln(x)}{x^2}dx.$$ I have tried substitution with $u=\ln(x)$, then $du = 1/x dx$, but this only takes care of one of the $x$ on the bottom: $$ \int \frac{u}{x} du. $$ I ...
7
votes
6answers
430 views

Find the following integral: $\int {{{1 + \sin x} \over {\cos x}}dx} $

My attempt: $\int {{{1 + \sin x} \over {\cos x}}dx} $, given : $u = \sin x$ I use the general rule: $\eqalign{ & \int {f(x)dx = \int {f\left[ {g(u)} \right]{{dx} \over {du}}du} } \cr ...
4
votes
2answers
112 views

Evaluate $\lim_{n \to \infty} \int_{0}^1 \frac{n+1}{2^{n+1}} \left(\frac{(t+1)^{n+1}-(1-t)^{n+1}}{t}\right) \mathrm{d}t$

Evaluate $$\lim_{n \to \infty} \int_{0}^1 \frac{n+1}{2^{n+1}} \left(\frac{(t+1)^{n+1}-(1-t)^{n+1}}{t}\right) \mathrm{d}t$$ For this integral, I have tried using integration by parts and then ...
2
votes
2answers
39 views

Integral by using substitution (How to proceed?)

Using the substitution $x=a\sin\theta$, or otherwise, find $\int\frac{1}{x^2\sqrt{a^2-x^2}}dx$. My attempt, $x=a\sin\theta$ $dx=a\cos (\theta)d\theta$. Then $\sqrt{a^2-x^2}=\sqrt{a^2-a^2\sin ...
0
votes
0answers
16 views

How do I find the mass of a circular disk?

In this question I have been given the specific weight of the disk as 15kN/m^3 and have the radius of the disk as 0.25m. how do I find the mass of the disk? (To then go on and find the moment of ...
1
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1answer
15 views

Determining Line Integrals from a Graph and Vector Field (Image Included)

Consider the vector field: $$F=\left(\frac{2xy-2xy^2}{\left(1+x^2\right)^2}+\frac{8}{13}\right)i+\left(\frac{2y-1}{1+x^2}+2y\right)j$$ Determine $$\int_C F\cdot dr$$ where $C$ is the path ...
0
votes
0answers
45 views
1
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0answers
12 views

Euler method global truncation error and conservation of orbital energy.

I've been given a simplified model of a small mass orbiting a much larger one, which I need to solve using Euler's method. I've reduced the equation to two (or four, really) coupled first order ODEs: ...
0
votes
0answers
39 views

Diffrental equation solution [on hold]

How can I solve this equation? $$\frac{\partial f}{\partial x} =\frac{a-x}{y} \frac{\partial f}{\partial y}$$ where $a$ is a constant. So what is $f$?
0
votes
0answers
8 views

if $F(s_{0})$ for some $s_{o}$exists then it exists for all $s>s_{o}$

if laplace transform $F(s_{0})$ for some $s_{o}$exists then it exists for all $s>s_{o}$. i need to prove this . now, ...
3
votes
1answer
44 views

Given $f\in L^1(\mathbb{R})$ with $\|f\|_1<\infty$ and $g_n=\sqrt{n/2\pi}e^{-nx^2/2},f_n=g_n\ast f$, show that $\lim\|f_n-f\|_1=0$

Given $f$ a Lebesgue integrable function on $\mathbb{R}$ with finite $L^1$-norm, I am asked to show that $\lim_{n\to\infty} \|f_n - f\|_1 = 0$, where $f_n = f \ast g_n$ and $g_n = ...
2
votes
1answer
68 views

Showing that $\int fg\le \int g$ implies $f=0$ a.e.

Take $0<p<1$. If $f$ is locally integrable over on $\mathbb{R}$ and $$\Bigg\vert \int fg\Bigg\vert\le \Vert g\Vert_p\tag 1$$ for every $g$ continuous on a set of compact support, then $f=0$ a.e. ...
5
votes
3answers
61 views

Let $1 \leq p <\infty$ and $f \in L^p(\mathbb{R})$. Prove $\lim_{x \to \infty} \int_x^{x+1} f(t) dt = 0$.

(Jones, p. 246) Let $1 \leq p <\infty$ and $f \in L^p(\mathbb{R})$. Prove $\lim_{x \to \infty} \int_x^{x+1} f(t) dt = 0$. This seems pretty easy to prove in the following way: Let $g_j$ be a ...
3
votes
1answer
55 views

Volume of a Solid of Revolution Rotated Around the Y-Axis

Sorry to post an obvious homework question here, but my daughter's calculus teacher isn't much on "teaching" and left a problem like this one out of the notes. I can't find much on the internet to ...
-1
votes
2answers
35 views

Integrating to find deceleration, and finding ball height? [on hold]

1) A ball is thrown straight up from a height of 8 ft with an initial velocity of 40 ft/sec. How high will the ball go? (Take g = 32 ft/sec2.) How would I do this? Wouldn't I need to find a velocity ...
0
votes
1answer
14 views

Lower semicontinuous non-negative function on a locally compact Hausdroff space with a countable base

An extended real number is an element of $\mathbb R \cup \{-\infty, +\infty\}$. Let $X$ be a locally compact Hausdorff space with a countable base. An extended real valued function $f$ on $X$ is ...
1
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2answers
48 views

Antiderivative of $ (x^2 + c)^{-3/2} $ [on hold]

What method should be used to determine the antiderivative of this expression? Edit: I have $ c > 0 $ in the problem I'm working on.
1
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0answers
41 views

The $L^p$ convergence rate of the tail of the series $\sum_{n=1}^{\infty}\min\{1,2^n |x|^{-1} \}2^{-na}$

This a follow-up to the question: Convergence Rate of the Tail of the Series $m^{a}\sum_{n=1}^{\infty}\min\{1,2^n m^{-1} \}2^{-ja}$ When $a > 0$, we have $$ \sum_{n=1}^{\infty}\min\{1,2^n |x|^{-1} ...
0
votes
0answers
28 views

Any hint for solving this Poisson's integral? [on hold]

I tried various approach without success in solving this integral: $\frac{1}{2\sqrt{\pi t}}\int_{\mathbb{R}} e^{\frac{-(x-y)^2}{4t}}\phi (y) dy$ which is the solution to the heat equation. I only have ...
2
votes
1answer
90 views

Bonus integration problem we got at class: Integrate $\frac {x \sin x}{1+\cos^2x}$ between $0$ and $\pi$ [duplicate]

Bonus integration problem we got at class: Integrate $\frac {x \sin x}{1+\cos^2x}$ between $0$ and $\pi$ So the lecturer gave this problem. I tried this really hard but couldn't succeed. It ...
2
votes
1answer
43 views

Integration over ellipse

$A=\{(x,y)\in \Bbb R^2\mid \frac{x^2}{a^2}+\frac {y^2}{b^2}=1\}$. Find $\int_A (\cos x)y\,dx+(x+\sin x)\,dy$. Can someone please please give a methodological answer? Thanks a lot!
0
votes
0answers
19 views

A question of multi-dimensional integral

Consider the function $$\Omega(N,E)=\int dE_1 \int dE_2 \cdots \int dE_N \Omega_1(E_1)\Omega_2(E_2) \cdots \Omega_N(E_N)\delta(E-E_1-E_2\cdots -E_N)$$ Is there a sufficiently condition on the ...
1
vote
1answer
38 views

Existence of double integral

the short time Fourier transform is obtained by the formula: $$Sf(u,\epsilon)=\int_\mathbb{R}f(t)g(t-u)e^{-i\epsilon t}dt$$ where $f,g \in L^2(\mathbb{R})$ are the signal and window respectively: ...
2
votes
3answers
155 views

How to solve this integral by a simple way?

I'm given $$\int \frac{x^3}{\sqrt{x^4+x^2+1}}dx$$ My attempt, Let $u=x^2$, $du=2xdx$ $$=\frac{1}{2}\int \frac{u}{\sqrt{u^2+u+1}}du = \frac{1}{2}\int ...
12
votes
0answers
206 views

Integrate this monster

Can you please help me? I've been trying for some time now to integrate this: $$\int_0^\infty g^{-(a+1)} \; \exp\left\{-\left(\frac{b}{g} + \frac{1}{2} \sum_{i=1}^{n} ...
0
votes
3answers
72 views

Integrate $\frac{1}{1+\cos^2x}$. Probably need using some trigonometric identity I don't know

Integrate $\frac{1}{1+\cos^2x}$ I probably need using some trigonometric identity I don't know. I tried all methods I'm familiar with. Any assistance will be great. Thank you!
1
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2answers
54 views

How does the first fundamental theorem of calculus guarantee the existence of antiderivatives of functions?

First fundamental theorem of calculus: $$g(x) = \int_a^xf(t)dt$$ then $$g'(x) = f(x)$$ But how does this guarantee the existence of antiderivatives of functions? Tutorials always state it does, but ...
3
votes
3answers
66 views

Integral of trig fraction using substitution?

I'm chewing on an integral problem and don't have a clue where to begin. If someone could assist by suggesting a good starting point, I'd really appreciate it! Not asking for anyone to solve the ...
0
votes
2answers
36 views

Evaluating a complex integral (Hints please)

I am supposed to be able to show that, given $f(z)=\frac{1}{\pi}\int_0^1r\int_{-\pi}^\pi\frac{d\theta}{re^{i\theta}+z}dr$ then $f(z)=\overline{z}$ for $|z|<1$ and $f(z)=1/z$ if $|z|\geq1$. (This ...
1
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0answers
39 views

Integration of a function of two variables

How can we check the integrability of $f$ defined on $[0,1] \times [0,1]$ as $f(x,y)=$\begin{cases} 0 & x=\frac{1}{2},y \in \mathbb Q \\ 1 & x=\frac{1}{2},y \in \mathbb Q^c \\ ...
0
votes
1answer
36 views

Congruent Sub-Intervals with Reimann-Integrable Functions

Let $f:[a,b]\to\Bbb R$ be a Riemann-integrable function. Prove that for each $\sigma\gt0$ there exists a partition $\mathcal P$ of $[a,b]$ into congruent sub-intervals(that is, $x_{j}=a+{j(b-a)\over ...
1
vote
1answer
39 views

Triple integral volume by equations

I have trouble setting up a triple integral to find volume bound by equations, such as: $$z = x^2 + 3;\quad y = 3 - x^2;\quad x + y = 2;\quad z = 0.$$ I'm not sure how to figure how to find the ...
2
votes
2answers
94 views

How to solve $\int \frac{(x-1)\sqrt{x^4+2x^3-x^2+2x+1}}{x^2(x+1)}dx$?

I need to compute $$\int \frac{(x-1)\sqrt{x^4+2x^3-x^2+2x+1}}{x^2(x+1)}\ dx.$$ I tried it on wolfram but it timed out, maybe because I am on a mobile device. Any hint is appreciated.
2
votes
0answers
31 views

If $A$ and $f$ are bounded, then $f$ is integrable in the extended sense (?) [Spivak]

I have a problem with one of the theorems in Spivak's Calculus on Manifolds. I will give some background first: An open cover $\mathcal{O}$ of an open set $A \subset \mathbb{R}^n$ is admissible if ...
1
vote
1answer
26 views

Joint density function problem

I have a joint density function of Random Variables X and Y given by: $$ f(x,y) = \begin{cases} 2e^{-x}e^{-2y} & 0<x<\infty, 0<y<\infty \\ 0 &\text{otherwise} \end{cases} $$ And ...
0
votes
0answers
21 views

Correctly setting up flux integrals

My question has to do with picking the correct limits for integration. I thought I had it figured out well, but I had an interesting issue with a homework problem. The problems were about Green's ...
2
votes
4answers
104 views

Compute $\int _{\frac{4}{5}}^2\:f^{-1}\left(x\right)dx$

We have to compute $\int _{\frac{4}{5}}^2\:f^{-1}\left(x\right)dx$ where $f\left(x\right)=\frac{-x^3+2x^2-5x+8}{x^2+4},\:x\in \mathbb{R}$ is an bijective function. How help if we kno![enter image ...
1
vote
3answers
59 views

How to find $p(t)$ when $m$ varies linearly with $t$? [closed]

I have a function $p(t)$ (position and time) defined by $$p(t) = \frac{1}{2} \cdot \frac{F}{m} \cdot t^2$$ when the mass is constant. This is derived from Newtons second law and by integration of the ...
1
vote
2answers
87 views

$f(x)$ is Riemann integrable $\Rightarrow$ $\frac{1}{1 + f^2(x)}$ is Riemann integrable

Let f(x) be Riemann integrable on [a,b]. Then there exist $\lim_{x \rightarrow a+0} f(x)$ and $\lim_{x \rightarrow b-0} f(x)$ f(x) has only removable or jump discontinuities. The set of ...