Questions tagged [integration]
For questions about the properties of integrals. Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that describe the type of integral being considered. This tag often goes along with the (calculus) tag.
69,923
questions
-1
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Integral of the square of a floor function in an infinite sum
Consider the given integral:
$$
\int_{0}^{1} (\sum_{n=1}^{\infty}\frac{\lfloor 2^n x\rfloor}{3^n})^2 \,dx
$$
Please suggest how to approach this problem.
- 25
0
votes
0
answers
16
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On the commutativity of integral and limit
Suppose that $f\in C^1$
I try to prove that $\frac{d}{dt}\int_{0}^{t}h(t,x)dx=h(t,t)+\int_{0}^{t}\frac{\partial h}{\partial t}(t,x)dx$.
However this depends on the fact that $\lim_{\Delta t\to0} \...
- 31
-5
votes
0
answers
15
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Integrate ln|sinx|cos(2nx)dx from 0 to 2pie/3 , n belong to N [closed]
[enter image description here][1]
https://i.stack.imgur.com/XpYv8.png
0
votes
0
answers
37
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Shape differential forms
I'm reading "Differential forms in algebraic topology" by Bott. In chapter 1 they introduce differential forms as elements of
$\Omega^*(\mathbb{R})=\{C^{\infty}$ functions on $\mathbb{R^n}\}\...
1
vote
0
answers
36
views
How to derive elliptic integral of the first kind from $\int_{-\pi}^{\pi}\frac{1}{\sqrt{(\eta^{2}/2 - 3/2 -\cos 2\theta)^2+\cos^2\theta}}\ d\theta$ [duplicate]
Spent several days already trying to figure out how to convert the given integral to this: $$\frac{8}{\sqrt{(\eta-1)^3(\eta+3)}}K\left(\sqrt{\frac{16\eta}{(\eta-1)^3(\eta+3)}}\right)$$
where
$$
K\left(...
- 21
2
votes
1
answer
26
views
Pettis integral on locally convex space and seminorms
Let $E$ be a locally convex Hausdorff space, and $X$ be a locally compact Hausdorff space which we fix a positive Radon measure $\mu$. Assume that $f: X \to E$ is a function such that the Pettis-...
- 894
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votes
0
answers
40
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Integration. Solution verification - I need to know if my thinking is correct
Let $\Omega$ be a subset of $\mathbb{R}^2$, a convex region whose boundary $\partial\Omega$ is the image of a 1-periodic function $\gamma: \mathbb{R} \to \mathbb{R}^2$ of class $C^2$. Assume that $|\...
- 363
0
votes
0
answers
23
views
Dealing with an interval within the limit of integration
I am working through my convolution homework for a signal processing class and I am very perplexed by one exercise in particular. It involves integrating a rectangular signal $x(t) = u(t) - u(t-2)$ by ...
- 1
1
vote
0
answers
48
views
How is proved that for $a\in\mathbb{R}$ is true that $\displaystyle{\int} x(t)\delta(t-a)\ dt = x(a)\ \theta(t-a)+\mathbf{C}$?
How is proved that for $a\in\mathbb{R}$ is true that $\displaystyle{\int} x(t)\delta(t-a)\ dt = x(a)\ \theta(t-a)+\mathbf{C}$?
Here $\mathbf{C}$ is the integration constant for the indefinite integral,...
- 1,183
3
votes
1
answer
92
views
Doubts in $\lim_{x \to +\infty} \frac{\int_{-x}^x \frac{1}{y^2}dy}{x}$
Consider the limit:
$$\lim_{x \to +\infty} \frac{\int_{-x}^x \frac{1}{y^2}dy}{x}$$
Since $\int_{-x}^x \frac{1}{y^2}dy=\int_0^x \frac{1}{y^2}dy-\int_0^{-x} \frac{1}{y^2}dy$, using Hopital's rule:
$$\...
- 799
1
vote
1
answer
38
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If $f'(t)=x'(t)\ \theta(T-t)$, What is the accurate formulation of $f(t)$? (antiderivative over distributions/special_functions)
If $f'(t)=x'(t)\ \theta(T-t)$, What is the accurate formulation of $f(t)$? (antiderivative over distributions/special_functions)
Maybe the question look simple, by I need to understand Why it is as so,...
- 1,183
1
vote
0
answers
72
views
$\sum \geq \int \mathrm{ or } \sum \leq \int?$
If I have a function that is continuous, differentiable and positive, then would its sum be greater or its integral? Or is there no fixed answer? If so, under what factors is what greater?
$$\sum_{x=1}...
- 320
0
votes
0
answers
62
views
How do we calculate the following integral $\int_{0}^{1} e^{tx}\left[\frac{1 - \cos(2x)}{2x}\right]\mathrm{d}x$? [closed]
For $t>0$, compute $$\int_0^1 e^{t x} \frac{1-\cos 2x}{2x}\,dx$$
Is using the exponential integral the only way to compute this?
- 19
0
votes
0
answers
62
views
Integral of convolution squared
Given two integrable functions, we know that
$$ \int_{-\infty}^{\infty} (f*g)(x)dx = \bigg(\int_{-\infty}^{\infty}f(x)dx\bigg) \bigg(\int_{-\infty}^{\infty}g(x)dx\bigg) $$
So we can recover the ...
-2
votes
0
answers
47
views
Difficult integral in statistical mechanics. [closed]
Can someone explain how I should approach solving the integral displayed in the image (4.10)? The integral is solving for the average velocity orthogonal to region in the second picture. We are only ...
-3
votes
0
answers
17
views
2
votes
1
answer
65
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How is a Hankel contour different from a keyhole contour?
From what I'm guessing a keyhole contour is one that looks like this
and because it can be shown that the contribution from $C_R$ and $C_\epsilon$ vanishes as $R\to \infty$, a Hankel contour looks ...
- 151
0
votes
0
answers
71
views
How can I get the following estimate?
I am having a hard time getting the following lower estimate. Is it obvious? At first, I thought it was, but now I do not see it that way, so any hint would be appreciated.
$$\frac{\int_{M}|\nabla \...
- 47
0
votes
0
answers
30
views
Saddle Point analysis of integral
I want to make saddle point approximation of the integral $$ I = \frac{1}{2 \pi i } \int_{\gamma - i \infty} ^{\gamma + i \infty} \frac{1}{z^6} e^{zt} dz $$ but as I see the function in the exponent i....
0
votes
0
answers
30
views
Help with Calculus Problem: Finding Maximum Value of Integral [closed]
I'm currently struggling with a calculus problem and would appreciate any help or guidance. The problem is as follows:
The figure shows the graphs of the functions $F$ and $G$. For the functions $f$ ...
- 45
2
votes
1
answer
72
views
How to prove the limits about integral. [closed]
Let $f (x)$ be continuous on $[0,a](0<a<\pi)$.prove that
$$\lim_{\lambda\to+\infty}\frac{1}{\lambda}\int_{0}^{a}f(x)\frac{\sin^2\lambda x}{\sin^2x}\mathrm{d}x = \frac{\pi}{2}f(0).$$
I thought ...
- 1,149
0
votes
2
answers
36
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Integrating w.r.t. two different variables - a physics problem on the maximum height of a vertically thrown object
I'm trying to derive an equation for the height an object reaches after being vertically thrown upwards, where the two forces acting on it are gravity and quadratic air resistance, so that
$$
ma=-mg-...
2
votes
1
answer
59
views
Applying Fundamental Theorem of Calculus to improper integral
Let's consider $f:~]-\infty,\infty[~\to\mathbb{R}$ a continuous function. Our professor nonchalantly said that if we assume that $\int\limits_{-\infty}^xf(t)dt$ exists, then differentiating by using ...
- 3,685
0
votes
1
answer
51
views
Dealing with the $dx$ that is in the denominator of a definite integral.
I’m not sure if I should specify between a definite or an indefinite integral here but that’s the instance I’m specifically interested in.
I was thinking about an integration rule that states that if ...
- 190
0
votes
0
answers
27
views
Is the derivative of the exponential family's density function absolute integrable?
Consider the exponential family of distributions, whose density can be written as:
$$f_\theta(x)=h(x)\exp\left[\eta(\theta)^T T(x)-A(\theta)\right]$$
where $h: \mathbb{R}\to \mathbb{R}^+$, $\theta$ is ...
- 89
0
votes
0
answers
31
views
Exchanging integration order and changing limits of integration with Dirac delta
I was wondering whether it'd be possible to change the order of integration in this case and whether I'm doing it correctly.
The integral is
$$\int_{0}^{\infty} x \int_{-1}^{1} \delta(x-\cos{\theta}) ...
-5
votes
0
answers
27
views
-1
votes
0
answers
74
views
Double Integral Evaluation over $[0,1]$
How do I evaluate the following integral.
$$\int_0^1 \int_0^1 e^{tx} \sin(2xy) dxdy, \ t>0$$
So far using Kurt's advice, I get:
$$\int_0^1 \int_0^1 e^{tx} \sin(2xy) dx dy = \int_0^1 \left(\int_0^1 ...
- 19
0
votes
1
answer
37
views
How to compute an integral over a sphere?
Let $r > 0$ then how to compute an integral $\int_{\partial B(0,r)}|x|^{-n}dS(x)$ where $S(x)$ is a standard spherical measure. I tried to move to polar coordninates, but i have difficulties with ...
0
votes
0
answers
41
views
How to solve this integral involving an ugly root in the denominator
I came across a crazy integration question involving polynomials
I tried making a substitution like ${\sqrt(x+1)}=t$ with a (good) intention as algebraic calculations could be easier without roots and ...
0
votes
0
answers
9
views
integral of power of gaussian CDF function multiplied by exponential
I am trying to solve the following integration (closed-form solution) or numerically:
$$
\int_{0}^{x}\left[1-erfc \left(\frac {x-\alpha}{sqrt(2) \sigma}\right)\right]^n \lambda e^{-\lambda \alpha} d\...
3
votes
2
answers
97
views
How can I solve $\int_{0}^{\infty}{\frac{\log(x)}{x^2+4}}dx$ using $2\tan\theta$ as substitution for x? [duplicate]
$$\int_{0}^{\infty}{\frac{\log(x)}{x^2+4}}dx$$
-from the book Advanced Problems in Mathematics by Vikas Gupta
So I tried to solve this integral by substituting $x=2\tan\theta$ so that we get the upper ...
-1
votes
0
answers
35
views
In the last line for the answer, the two values should be added right? (31/631 - 241/631 should be 31/631 + 241/631) [closed]
Question
Answer given
I've attached a picture with question and the given answer
-2
votes
1
answer
40
views
Calculating the maximum height a person can jump on the moon, given the maximum height they jumped on earth [closed]
A person on earth performs a high jump of 2.41m. Assuming the person is wearing an ideal spacesuit that does not adversely affect athletic performance, the acceleration due to gravity on the moon is 1....
1
vote
0
answers
30
views
Limit of $\sum_{t=0}^n a_n(t)$ for n tends to infinity
I am wondering about how to deal with a limit of the following form
$$ \lim_{n \rightarrow \infty} \sum_{t=0}^n a_n(t),$$
with the function $a_n(t)$ and the sum being dependent on $n$.
Consider the ...
- 33
0
votes
0
answers
44
views
Numerical Integration: Why isn't Polynomial Approximation Working?
I have the following integration problem:
$$ \int_0^1{ -m f(x) \left(\int_0^x{f(u)} du \right)}^{m-1} dx $$
I attempted to approximate $ \int_0^x{f(u)} du $ using Chebyshev interpolation, I took $n+1$...
- 13
0
votes
0
answers
46
views
Why does this Anti-derivative not give me the right answer? [closed]
I took the integral of 1/x
Now when I take the definite integral of $f(x)$ from $0$ to $\pi$ I get $0$ since $\cos(0)=\sin(\pi)=\arcos(4)=0$ however this does not match with the graph and online ...
0
votes
1
answer
41
views
Proving that two integral are equal (changing variable of integration) [closed]
I want to prove that
$$
\frac{1}{\beta^2}\int_0^\beta d\tilde{\beta} \frac{n^2 \tilde{\beta}}{e^{\tilde{\beta} n}+1}=-\int_{n}^\infty dz \frac{z}{e^{\beta z}+1}
$$
how do we change the variable of ...
- 215
1
vote
0
answers
53
views
How to calculate integral $\mathbb E[\log X]$?
I'm trying to calculate $\mathbb E[\log X]$ for X a non-negative random variable with finite mean, and
$$\mathbb E[\log X|X\leq t]=\frac{1}{F(t)}\int_0^t \log xf(x)dx=\frac{1}{F(t)}([\log zF(z)]^t_0-\...
- 47
1
vote
1
answer
69
views
Why can you integrate on both sides of an equation?
I have seen that you can integrate on both sides of an equation but that can't differentiate both sides of an equation. For differentiation it is easy to find examples why it doesn't work, but I haven'...
- 19
6
votes
5
answers
298
views
How to prove this asymptotic formula:$J_n=\int_{0}^{1}(1-x^2+x^3)^ndx\sim \sqrt{\frac{\pi}{4n}}$
How to prove this asymptotic formula:$$J_n=\int_{0}^{1}(1-x^2+x^3)^ndx\sim \sqrt{\frac{\pi}{4n}}$$
My idea is to solve this $J_n=\int_{0}^{1}(1-x^2+x^3)^ndx$ .Is it feasible to calculate its recursion ...
- 63
0
votes
0
answers
24
views
Expectation of two default times between a period
The setup:
Let two default times be defined as
$$\tau_i=\inf\left\{t: \int_0^t\lambda_i(s) \,ds \geq E_1\right\},\quad i\in\{1,2\}$$
Where $E_1$ is a standard exponential random variable and $\...
- 36
0
votes
1
answer
51
views
Fourier transform of $x \in \mathbb{R} \mapsto (1+x^2)^{-\alpha}$
I hope that the answer is not somewhere in the forum. I searched without success.
I would like to compute the fourier transform of $f:x \in \mathbb{R} \mapsto (1+x^2)^{-\alpha}$ where $\alpha > 1/2$...
- 1,996
0
votes
0
answers
49
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proving an expression involving integration
Sorry If this question does not belong here, I am trying for the last six days to prove the following integration but I am failing, and I will really appreciate a lit bit of help.
I have an expression ...
- 215
1
vote
0
answers
40
views
Line integral with $dx$ and $dy$ - geometric interpretation
This question is about curvilinear integration.
The first page here, it says what the geometric meaning is when the line integral is taken with this $ds$, where this $ds$ indicates the line integral ...
- 12.1k
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votes
0
answers
15
views
Why is it okay to swap variables in this integration exercise? [duplicate]
This is related to integration and variable substitution.
I'm having trouble understanding the official solution to the following exercise from Stewart Calculus:
If $f$ is continuous on $\mathbb{R}$, ...
0
votes
0
answers
17
views
$ \frac{1}{t_n} \int_0^{t_n} e^{2 \pi i h f(x)} \, dx \to 0$ if $ \frac{1}{n} \int_0^n e^{2 \pi i h f(x)} \, dx \to 0 $
Suppose that $f$ is a measurable functions on the real line and $h \in \mathbb Z \setminus \{ 0 \}$ such that
$$
\frac{1}{n} \int_0^n e^{2 \pi i h f(x)} \, dx \to 0
$$
as $n \in \mathbb N$ tends to ...
- 67
5
votes
2
answers
178
views
How to solve the following limit
If
$$\lim_{x\to0}\frac1{x^m}\prod_{k=1}^n \int_0^x\big[k-\cos(kt)\big]\mathrm dt$$
exists and is equal to $20$ (where $m,n\in\mathbb N$) then what is the value of $n$?
I started this question with ...
- 91
4
votes
1
answer
52
views
If $x,y∈(-π,π]$, then find the area of the polygon formed by points $(x,y)$ satisfying the equation $\lfloor|\sin x|\rfloor+\lfloor|\cos y|\rfloor=2$.
If $x,y∈(-π,π]$, then find the area of the polygon formed by points $(x,y)$ satisfying the equation $\lfloor|\sin x|\rfloor+\lfloor|\cos y|\rfloor=2$.
My attempts include using a graphing tool and ...
- 91
1
vote
0
answers
58
views
Integrals with massive amount of cancellation (positive and negative portions of integrand cancel out almost perfectly)
In Gamelin's Complex Analysis, there are exercises/examples (pg. 201-202) of the form
$$\int_{-\infty}^\infty \frac{P(x)}{Q(x)} \cos(ax) dx$$
The first "trick" is to use complex analysis: ...
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