Questions tagged [improper-integrals]
Questions involving improper integrals, defined as the limit of a definite integral as an endpoint of the interval of integration approaches either a specified real number or $\infty$ or $-\infty$, or as both endpoints approach limits.
7,617
questions
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Challenging integral transformation [closed]
$$\int_{0}^{\frac{\pi }{2}}\frac{\cos^2(\pi \tan x \sin x)\Gamma(\frac{1}{2}+\tan x\sin x) }{\Gamma(1+\tan x\sin x)}+\frac{\sin(2\pi \tan x \sin x)\Gamma(\tan x\sin x) }{2\Gamma(\frac{1}{2}+\tan x\sin ...
- 1
1
vote
1
answer
54
views
Closed form for an improper integral
Is it possible to further simplify the following improper integral?
$$
\int_{0}^{\infty}\frac{\ln(e^{x+s}+1)-\ln(e^{x}+1)}{se^{s}}ds,\;x>0
$$
The denominator leads me to think about Gamma function, ...
- 357
1
vote
0
answers
55
views
$\int_0^\infty \frac{\ln x}{(1+e^{x})^2}dx$ by contour integration
How would one compute the following using a contour integral in the complex plane? I know how to perform this integral by other means such as differentiation under the integration and series expansion....
- 9,593
0
votes
0
answers
10
views
For what values of $k$, $t_{0}$, or $x$ is this integral convergent?
$$I(x)=\int_{-\infty}^{0}{({-\xi})^{1.5k}\sin{(\xi{t_{0})}}e^{-i\xi{x}}}d\xi,$$
$k$ is an integer greater or equal to $0$, $t_{0}\in[0,\infty)$, $x\in(-\infty,\infty)$.
I need to find an approximate ...
0
votes
0
answers
66
views
Improper Integral of $\sqrt{\tan{x}}dx$ [duplicate]
I've just started studying improper integrals and I've come across this one:
$$\int_{0}^{\infty}{\sqrt{\tan{x}}dx}$$
The main issue I have with this is that the root is imaginary when tanx is negative,...
- 114
0
votes
1
answer
115
views
Is there an easy way to calculate this infinite summation
Is there an easy way to calculate this summation of integral:
$$\sum_{n=0}^\infty \int_{r=0}^1 \frac {(r-\frac{1}{2})\cos(c\cdot\ln(r+n))} {(r+n)^{1-b}} dr $$
The most obvious approach is to calculate ...
0
votes
0
answers
25
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Determining the Convergence of Improper Integral with Given Limit
Let $f(x)$ be a continuous function defined for $x \geq 0$ such that $f(x) \geq 0$ for all $x \geq 0$. Additionally, suppose that
$$ \lim_{x\to\infty} f(x) =a$$
where $a$ is a finite constant.
Is this ...
0
votes
1
answer
38
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Calculate a limit by Riemann lemma
This is a question I met in Fourier analysis chapter in the Mathematic Analysis.
Question. Calculate the limit $$\operatorname*{lim}_{\lambda\to\infty}\int_{0}^{1}\mathrm{ln}x\cos^{2}(\lambda x)\...
2
votes
1
answer
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$\int_{1}^{\infty} xf'(x) \space dx$ converges absolutely. Prove\disprove: $\int_{1}^{\infty} f(\xi) \space d\xi$ converges absolutely.
Let $f:[1, \infty) \rightarrow \mathbb{R}$ s.t $f$ is differentiable at $[1, \infty)$ and $\lim_{x \to \infty} f(x) = 0.$
Suppose that $f'$ is Riemann-integrable and the improper integral $\int_{1}^{\...
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4
votes
1
answer
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Challenge: $\int_0^\infty\frac1{1+x}\ln\left(\prod_{k=0}^\infty(1+e^{-(2k+1)\sqrt{x}})(1+e^{-(2k+1)\pi^2\sqrt{x}})\right)\,dx=\pi^3/24+\pi/24-\ln^22$
I have created a challenging integral problem, and I dedicate this challenge to fellow integral enthusiasts, inviting them to solve it
$$\small{\int_{0}^{\infty }\frac{\ln\left[\left ( 1+e^{-\sqrt{x}} ...
- 444
-2
votes
0
answers
50
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Improper integral confusion
I want to understand a solved exercise our professor gave us. If $f:[0,+\infty)\to[0,+\infty)$ is continuous and the improper integral (from zero to infinity) of $f(x)$ is equal to infinity: $\int_{0}...
0
votes
0
answers
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Is it possible to determine whether or not an improper integral between two assymptotes will converge or not, without having to find its primitive?
I was evaluating $\int_{-1}^{1}\frac{dx}{1-x^2}$ and came to the conclusion it does not converge. I understand it is because its primitive, $F(x)=\frac{1}{2}[\ln(1+x)-\ln(1-x)]$, does not converge at ...
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0
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0
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35
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Doubt regarding splitting up the interval of integration
Consider a definite integral function $f(x)$. We know that:
$\int_{a}^{b}f(x) dx= \int_{a}^{c}f(x) dx + \int_{c}^{b}f(x) dx$
I have a couple of conceptual doubts regarding this splitting of interval ...
7
votes
2
answers
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Can we evaluate the integral $ I(a)=\int_0^{\infty} \frac{\sin x}{x^a} e^{-x} d x, $ without Gamma functions?
Encountering the integral in the post stating that
$$
\int_{0}^{\infty} \frac{e^{-x} \sin(x)}{x}\,dx=\frac\pi4,
$$
I started to investigate the integral in a more general form as
$$
I(a)=\int_0^{\...
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1
vote
0
answers
17
views
Determining the convergence of improper integral
I would like to prove the convergence of the following improper integral:
$$\int_1^2 {\frac{\sqrt{1+x^2}}{\sqrt[3]{16-x^4}}} dx\quad\quad $$
I tried to find antiderivative (with assistance of online ...
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0
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How to solve Improper integrals by contour integration [duplicate]
I'm trying to solve the following integral
$$\int_{-\infty}^{\infty}\frac{\cos(2πx)}{x^4+4}$$
But I don't know how to approach it, could someone help me?
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votes
0
answers
45
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Path integrals with real integrands [closed]
I've cleaned up the actual formula. See below.
The usual Feynman path integral, on an interval [a,b], dividing [a,b] into N+1 equal seqments [a=xN, xN-1], …, [x1,b=x0], is defined as K(b,a) = lim(N→∞) ...
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votes
2
answers
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Can you give me an example of function $f$'s improper integral converges but $f\cdot f$'s integral doesn't converge?
Could you provide me with an example: a function $f$ is a positive continuous function, and the integral of $f$ from $a$ to positive infinity converges, but the integral of the square of $f$ from a to ...
- 1
4
votes
3
answers
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Solving $\int_0^\infty \frac{\sin x(1-\cos x)}{x^2}dx$
So we have the following improper integral and we have to prove that it converges(if it converges then find its value). $$\int_0^\infty \frac{\sin x(1-\cos x)}{x^2}dx$$
I tried the feynman approach of ...
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0
votes
0
answers
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I need to compute an asymptotic approximation to this improper integral using the method of stationary phase.
I need to compute an asymptotic approximation to this improper integral using the method of stationary phase.
$$I(x)=\int_{-\infty}^{0}{({-\xi})^{1.5k}\sin{(\xi{t_{0})}}e^{-i\xi{x}}}d\xi,\:\:\:\text{...
1
vote
1
answer
37
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Evaluating $\int_{p} f(p) \delta(p+a) \Theta(p)= f(-a) \Theta(-a)$ for Heaviside theta function
I want to evaluate the following integral
For given arbitrary function $f$ and Dirac Delta function $\delta$ with Heaviside Theta function $\Theta$, what is the form of
\begin{align}
\int_{p} f(p) \...
- 6,326
1
vote
0
answers
69
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A tricky integral problem depending on a parameter
I suggest directly see "my confusion" part and the last line of my solution to get a quick understand to save your time.
When I was learning the improper integral depending on a parameter ...
1
vote
1
answer
29
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About second order linear differencial equation with second term
I'm reading a physic paper with some math in it. I'm struggling following the author's mathematical equalities about differential equation. I explain:
Let $p$ a solution of
$$
p''\left(x\right) - k^2 ...
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votes
1
answer
35
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Is this reasoning for calculating this particular integral sound?
I've come across this function:
$ f(x) = (-2)^n, x\in\left [ n, n+1 \right ), n\in \mathbb{Z} $
and I was trying to calculate its integral going from 0 to infinity. Since you can't just take the ...
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0
votes
1
answer
44
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How to change integral bounds when calculating an improper integral
So it is estabilished that for definite integrals, the following holds:
$$\int_a^b f(g(x))g'(x)dx = \int_{g(a)}^{g(b)} f(u)du$$
given
$ u=g(x)$.
However, when calculating the expectation of a standard ...
0
votes
0
answers
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Integral Formula Involving Legendre Polynomials
The following exercise takes the form;
$\int_{0}^{\infty}f\left(\frac{P_{n+1}\left(x\right)}{P_{n}\left(x\right)}\right)\cdot\frac{1}{P_{n}\left(x\right)^{2}}dx=\left(n+1\right)\int_{0}^{\infty}f\left(...
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1
answer
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Fixed point iteration involving integral $f(x)=\int_{-\infty}^{\infty}k(x,t)f(x-t)dt+g(x)$.
Let $g(x)$ and $k(x,t)$ be two known functions. Is it possible to uniquely solve for $f(x)$ that is defined as:
$$f(x)=\int_{-\infty}^{\infty}k(x,t)f(x-t)dt+g(x) \tag{1}$$
I am looking for a numerical ...
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1
vote
0
answers
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Why not expand the Lebesgue integral by summing up a series of multiple Lebesgue integrals? [duplicate]
Below is a classic example of an integral expression that can be computed using the improper Riemann integral, but not using the Lebesgue integral:
$$\int\limits_0^\infty\frac{sin(x)}{x}$$
The problem ...
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votes
2
answers
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Area between $\frac{\sin(x)}{x}$ and its derivative
I want to find the area of the space between $\frac{\sin{x}}{x}$ and its derivative, $\frac{x\cos(x)-\sin(x)}{x^2}$, that includes the origin. More specifically, between the sinc function and its ...
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1
vote
0
answers
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views
What is the best way to numerically integrate this function?
I am trying to numerically evaluate an integral that came up in my work. The integral takes the form
$$
\int_0^{\infty}dx\tanh(\alpha(x-x^*))\int_0^{\infty}dzf(x,z)\delta(g(x,z)-\beta t(z))
$$
with $\...
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2
votes
0
answers
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Existence of the limit $\lim_{R\to\infty} \int_0^R \frac{\sin x}{x}\; dx$
I must prove the existence of the limit $$\lim_{R\to\infty} \int_0^R \frac{\sin x}{x}\; dx$$ The justification given is as follows: for all $R>\frac{\pi}{2}$ we have $$\begin{align}\int_0 ^R\frac{\...
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votes
2
answers
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Calculating variance for normal distribution
A random variable is said to be normally distributed with parameters $\mu$ and $\sigma^2$, if its density function is given by $$f(x)=\frac{1}{\sqrt{2\pi}\sigma}e^{\frac{-(x-\mu)^2}{2\sigma^2}}\;\;\;-\...
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votes
2
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Limit Bessel Gaussian
I am able to prove that:
$$\lim_{\varepsilon \to 0}\int^{\infty}_{0}\varepsilon^{-1}\left(J_{\frac{3}{2}}\left(\frac{r}{\varepsilon}\right)\right)^{2}\exp{(-r^{2})}r dr<\infty.$$
But I am unable to ...
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0
answers
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Indefinite integral | How to compute the second-order partial derivatives of Mixed (2 gaussian & 2 binary) Gaussian Copula?
This is a problem about "Computing the Mixture Gaussian Copula with 2 normal (continuous) variables and 2 binary (discrete) variables".
Problem background
I have two continuous r.v. and two ...
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votes
0
answers
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Residue: $ \operatorname{Res}_{z=\infty} \exp\left[i\pi z + z \log z + (1-z)\log(1-z)\right] $ [closed]
I would like to calculate the following residues, but the calculation is difficult and I can't find the value:
$$ \operatorname{Res}_{z=\infty} \exp \left[i\pi z + z \log z + (1-z) \log (1-z)\right] $$...
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vote
0
answers
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Please check my proof that a weird-integral-like $R^2 \to R$ function is continuous
Let $f \in C^0_c(R)$ (continuous function with compact support) and $w \in L^1(R)$ (Lebesgue integrable function) such that $w(x) \geq 0$ for all $x \in R$. The running autocorrelation function (it is ...
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votes
2
answers
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Double Limits When Defining Improper Integral
In my textbook , an improper integral is defined as shown below, where $F$ is a primitive function of a function $f$.
$$\int_{-\infty}^{+\infty}f(x)\mathrm{\ d}x\mathrel{\overset{\text{def}}{=\mathrel{...
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votes
1
answer
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Confusion regarding indefinite integral
Consider the integral
$$
\int_0^1 \frac{dx}{x(3+\sin(\log(x))}.
$$
Since $$\int_0^1 \frac{dx}{x(3+\sin(\log(x))}>\int_0^1 \frac{dx}{4x}$$.
The considered integral is infinite.
However, if we ...
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1
vote
1
answer
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Proving that $\int_1^{\infty} \frac{e^{-2at}}{\sqrt{{t-\frac{1}{t}}}} dt = \sqrt{\frac{a}{\pi}}\,\mathrm{K}_{\frac34}(a) \,\mathrm{K}_{\frac14}(a)$
As the title says, I'm interested in the proof that
$$
\int_1^{\infty} \frac{e^{-2at}}{\sqrt{{t-\frac{1}{t}}}} dt = \sqrt{\frac{a}{\pi}}\,\mathrm{K}_{\frac34}(a) \,\mathrm{K}_{\frac14}(a)
$$
for $a&...
- 891
1
vote
2
answers
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Improper integral convergence implies the existence of an infinite series which its partial sum converges
Let $f:[0, \infty) \rightarrow \mathbb{R}$ be a Riemann-integrable function at $[0, \beta]$ for each $\beta \in (0, \infty)$.
Suppose that $\forall x \in [0, \infty): \space f(x) \geq 0$.
If the ...
- 877
2
votes
1
answer
76
views
Improper integrals divergence\convergence
Let $f:[0,\infty)\rightarrow[0,\infty)$ be a real differentiable function such that $f(1)=1$.
If $f'$ is continuous and $\,\forall x\in [1,\infty):\,f'(x)\leqslant-\frac{2}{x^3}$,
then the improper ...
- 877
4
votes
1
answer
65
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A variation of Dirichlet's test for improper integrals
Let $f, g: [a, \infty) \rightarrow \mathbb{R}$ be real functions
s.t $f$ is continues at $[a, \infty)$ and $\lim_{x\to\infty}f(x) = 0$.
If $g$ is Riemann-integrable at $[a, N] \space\space, \forall N \...
- 877
0
votes
1
answer
61
views
Calculate the area between $f(x) = x \cos^2(x)$ and $g(x) = x$
I want to calculate the area bounded by the curves $f(x) = x \cos^2(x)$ and $g(x) = x$.
To find the points of intersection, I set $f(x)= g(x)$.
$$\begin{aligned}x\cos^2(x)&= x\\
x\cos^2(x)-x&= ...
0
votes
1
answer
100
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(Dis)Prove that the two integrals are equal
(Dis)Prove that $$\int_0^{\infty}\frac{\sin^2x}{x^2}dx=\int_0^{\infty}\frac{\sin x}{x}dx$$
I don't know the approach for these kind of problems involving improper integrals. Like what should I do? ...
0
votes
1
answer
59
views
Calculating $\int_0^1 2x \sin(1/x^2 )-2/x \cos(1/x^2 ) dx$
Calcule the improper integral $$\int_0^1 2x \sin \left( \frac 1 {x^2} \right)-\frac 2 x \cdot\cos\left(\frac 1 {x^2} \right) \mathrm{d}x.$$
It is pretty easy to see that the integral convreges, ...
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6
votes
2
answers
139
views
Closed-form for $\int_0^\infty \frac{a}{x}\sqrt{x^2+\frac{a^2}{x^2}} \exp\left(-x^2-\frac{a^2}{x^2}\right)dx$
Let $a > 0$. Is there a closed-form for
$$
\int_0^\infty \frac{a}{x}\sqrt{x^2+\frac{a^2}{x^2}} e^{-x^2-\frac{a^2}{x^2}} dx \ ?
$$
I have searched a lot on this website and the literature, but the ...
- 891
0
votes
0
answers
34
views
Can an improper integral with infinity singularity converge for any integrand?
I have an integral $\int_0^\infty f(x) dx$, where $\lim_{x \rightarrow 0} f(x) = \infty$ and $\forall x \geq 0: f(x)>0$. Is there any $f(x)$ that fulfills these requirements such that the integral ...
- 21
3
votes
2
answers
261
views
Show $\frac{\pi^2}{6}\sqrt{ab+\sqrt{ab+2\sqrt{ab+3\sqrt{\cdots}}}}>4$ by hand .
Problem :
Show that if :
$$a=\int_{0}^{1}x!dx,b=\int_{0}^{\infty}1/\Gamma(x)dx,x!=\Gamma(x+1)$$
Then we have :
$$S=\frac{\pi^2}{6}\sqrt{ab+\sqrt{ab+2\sqrt{ab+3\sqrt{\cdots}}}}=4.0054\cdots>4$$
...
- 3,705
3
votes
2
answers
138
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How to deal with a substitution that makes both the limits/bounds of an integral tending to infinity?
My question might seem similar to this question -"https://math.stackexchange.com/questions/1965777/substitution-makes-the-integral-bounds-equal" but I'll expand on why it doesn't resolve my ...
- 75
7
votes
3
answers
237
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Estimating $\int_0^1 a^x\ \frac{x(1-x)}{\sin(\pi x)} dx$
I doubt a closed form exists, so I am trying to approximate the integral:
$$I(a)=\int_0^1 a^x\ \frac{x(1-x)}{\sin(\pi x)} dx$$
I am therefore looking for a function $F(a)$ that provides
$$F(a)\simeq ...
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