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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.

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Integrating a function composed with a Dirac delta [closed]

In a physics problem I found an integral of the form: $$ \int d^4x\exp\left[i\delta(t-\tau)\hat{A}\right], $$ where $\hat{A}$ is an hermitian operator, $\tau\in\mathbb{R}$ and $d^4x=dtd^3x$. The ...
Albus Black's user avatar
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52 views

Estimation of a gamma function-like integral

A random variable $X$ has a pdf: $$f(x) = \frac{1}{k!} \cdot x^k \cdot e^{-x}$$ Prove that $$P(0<X<2\cdot(k+1)) > \frac{k}{k+1}$$ There are no conditions about $k$, so it can be non-integer. ...
Disciple's user avatar
  • 305
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0 answers
26 views

For which values of the parameter $S$ is the integral convergent?

I feel like I should take lower limit and I did but I couldn't proceed. I need help. Should I use some comparison test 1 $$\int \limits^{2}_{1} \frac{(1- \cos \pi x)(x-1)^S}{x^3 -1} dx$$
Struggling Mathematician's user avatar
1 vote
2 answers
33 views

Laplace transform of $\sin(\omega t)$

I am learning about the Laplace transform and I know I got the answer to this example question wrong, but I'm trying to figure out if I just made a calculus or algebra type error, or if I'm ...
Keara's user avatar
  • 125
3 votes
1 answer
79 views

Can we convert the following integral equation to a differential equation:$h(r)= \int_0^\infty\frac{f(x)}{e^{r x} + 1} dx$?

Can we convert the following integral equation to a differential equation:$$h(r) =\int_0^\infty\frac{f(x)}{e^{r x} + 1 } dx?$$ Here, $f(x)$ is a non-trivial 'nice' function( whatsoever condition is ...
stephan's user avatar
  • 375
-3 votes
0 answers
14 views

Is the Riemann Liouville fractional integral compact operator? [closed]

I am about to figured out that is the Riemann Liouville fractional integral compact operator or not? where f is continuous function in [0, b].
Jabar S. Hassan's user avatar
0 votes
2 answers
52 views

Integral of a Generalized Laguerre Polynomial [closed]

I am looking for the solutions to the following integral: $$ I_{n} = \int_{0}^{\infty}x^{4} \operatorname{L}_{n}^{3}\left(x\right) {\rm e}^{-\left(n + 3\right)x/2}\,{\rm d}x,\qquad n \in\mathbb{N}_{0} ...
Rocky's user avatar
  • 19
2 votes
0 answers
130 views

Finding a general expression for the improper integral $\int_0^\infty K_1( ( k^2+\alpha^2)^{1/2}r)\sin(kz)\,\mathrm{d}k$

$\newcommand{\on}[1]{\operatorname{#1}}$ In solving a classical fluid mechanics problem involving flow in porous media, I encountered a delicate infinite integral ...
Siegfriedenberghofen's user avatar
1 vote
0 answers
57 views

How do I evaluate $\int_0^1 \int_0^\infty \frac{\ln t}{1 + \alpha t^2} \mathrm{d}t \, \mathrm{d}\alpha$? [closed]

How would I evaluate the integral $$ \int_0^1 \int_0^\infty \frac{\ln t}{1 + \alpha t^2}\, \mathrm{d}t \, \mathrm{d}\alpha? $$
ANNJEE's user avatar
  • 49
1 vote
0 answers
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An integral identity involving a generalized hypergeometric function.

Let $\theta \ge 0$, $S\ge 0$, $\zeta \ge 0$ and $x \ge 0$ be real numbers and let $q \ge 1$ be an integer. Then the following identity below holds true: \begin{equation} G_\zeta(x) := \int\limits_0^\...
Przemo's user avatar
  • 11.5k
4 votes
2 answers
260 views

Could we approximate $\int_0^1\frac{1}{x^4}dx$ using a Riemann sum?

We know that in one dimension, the integral $\int_{0}^{1}\left(1/x^{4}\right){\rm d}x$ is not finite. But could we approximate this integral using a Riemann Sum ?. ...
Chang's user avatar
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5 votes
1 answer
125 views

Prove the closed form of $\int_0^T \exp\left(\frac{ia}{T-\tau}+\frac{ib}{\tau}\right)\frac{d\tau}{\left[\sqrt{(T-\tau)\tau}\right]^3}$.

While working through the Dover book "Quantum Mechanics and Path Integrals", I stumbled across a problem requiring me to use the following identity given in the book's appendix. $$\int_0^T \...
Anne Jones's user avatar
2 votes
1 answer
94 views

How do I find an equivalent for this integral?

I am trying to derive concentration bounds for the spectral norm of some rank-one matrices under Gaussian measure. My objective is to obtain a bound with respect to both the number of samples $N$ and ...
ylefay's user avatar
  • 169
2 votes
3 answers
412 views

how to solve improper integral

$$\int^{a}_{l}\frac{a^3 - x^3}{(a^2 - x^2)^{3/2}} {\rm d}x$$ I have obviously separated the integrals into $$\int^{a}_{l}\frac{a^3}{(a^2 - x^2)^{3/2}} {\rm d}x$$ $$\int^{a}_{l}\frac{- x^3}{(a^2 - x^2)^...
Argentum's user avatar
1 vote
1 answer
41 views

Sufficient condition for integrability

Let $f : (a,b)\to\mathbb R$ be a continuous function. Is is integrable, in general? What if the domain was $[a,b]$? If it's only generally integrable in the second case, is it because of uniform ...
Elvis's user avatar
  • 610
3 votes
0 answers
46 views

Integral divergence implies summation divergence

Assume $f:(0,1)\times (0,1) \to \mathbb{R}$ is a nonnegative continiuos bivariate function such that $$\int_0^1 \int_0^1 f^2(x,y) dx dy = \infty,\quad \int_0^1 \int_0^1 f(x,y) dx dy = 1.$$ Can we ...
efsdfmo12's user avatar
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4 votes
2 answers
293 views

Is there a way of finding a closed-form expression for $\int_0^\infty\frac{k^2J_0(k)^2\,\mathrm{d}k}{k^4+\left( k^2+x^2 \right)^2}$, $x\in\mathbb{R}$?

I am trying to find a closed-form expression for the following improper integral $$ \int_0^\infty \frac{k^2 J_0(k)^2 \, \mathrm{d}k}{k^4 + \left( k^2 + x^2 \right)^2} \, , $$ where $x \in \mathbb{R}$....
Siegfriedenberghofen's user avatar
1 vote
0 answers
44 views

Check the Integration process for this integration: $\int_0^s \exp\left(-ax+\frac bx\right)~dx$

I am trying to integrate $\int_0^s \exp\left(-ax+\frac bx\right)~dx$. I proceed in this way (according to Primitive of $ \exp(-a \times x- \frac bx)$): $=\int_0^s \exp(-ax-(\frac {(-b)}{x})~)~dx$ $=s\...
Abhishek Ghosh's user avatar
0 votes
2 answers
61 views

Use Taylor expansions to find if an improper integral converges

I'm trying to decide if the following integral $$\begin{align*} \int _{1}^{\infty} \frac{(e^{1/x^{2}}-1)^{\alpha}}{\log^{\beta}\left( 1+ \frac{1}{x} \right)} \ dx \end{align*}$$ is convergent or not. ...
M_k's user avatar
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0 answers
72 views

Laplace transform of exponential functions with derivatives.

I have been trying to calculate the Laplace transform of these troublesome exponential functions: Having $\alpha \in \mathbb{R^+}$ 1.$\mathcal{L}\left\{e^{n \alpha t}\frac{f(t)}{t^2} \right\},n \in \...
Jmtz's user avatar
  • 98
2 votes
0 answers
20 views

For what values of $k$ does the double integral of $\frac{1}{\|x_A - x_B\|^k}$ over intersecting surfaces converge?

I am working on a problem related to defining a potential energy function based on the distance between two intersecting surfaces in Euclidean space. Specifically, I am considering the following ...
cheng's user avatar
  • 21
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0 answers
77 views

Prove that $\int_0^{+\infty} e^{-t} \sum_{n = 0}^{+\infty} \frac{a_n t^n}{n!} dt = \sum_{n=0}^{+\infty} a_n$ without absolute convergence

I am trying to solve the following problem: Let $(a_n)$ be a sequence of complex numbers such that $\sum a_n$ is convergent ( but not necessarily absolutely convergent ). $\displaystyle\mbox{Prove ...
Kataevskiy's user avatar
0 votes
1 answer
32 views

Show that $\int_0^{\frac{\pi}{2}}\frac{x^m}{\sin^n x}dx$ is convergent if and only if $n < 1 + m.$

Show that $\int_0^{\frac{\pi}{2}}\frac{x^m}{\sin^n x}dx$ is convergent if and only if $n < 1 + m.$ The solution given in the book is as follows: Let the given integral be $\int_0^{\frac{\pi}{2}}f(x)...
Thomas Finley's user avatar
0 votes
1 answer
62 views

"De-singularize" an improper integral

Consider the improper integral $$ \int_0^\pi \frac{dx}{\sqrt{(\cos(x)+1+a)(\cos(x)+1)}}, $$ for some $a>0$. Does anyone know of a change of variables substitution that would make this integral ...
Gateau au fromage's user avatar
-1 votes
0 answers
79 views

How to prove $\int^\infty_0f(x)\frac{sin^2x}{x^2}dx=\int^{\frac{\pi}{2}}_0f(x)dx$? [duplicate]

Assumptions:f is even, integrable in $[0,\pi]$ and periodic with period $\pi$. Here is my attempt: Since $\int^\infty_0\frac{sin^2x}{x^2}dx=\frac{\pi}{2}$, we have $\int^\infty_0f(x)\frac{sin^2x}{x^2}...
user1337700's user avatar
1 vote
1 answer
105 views

Examine the convergence of $\int_0^1x^{n-1}\log x dx.$

Examine the convergence of $\int_0^1x^{n-1}\log x dx.$ The solution given is as follows: $0$ is the only point of infinite discontinuity of the integrand. Let us examine the convergence of $\int_0^{\...
Thomas Finley's user avatar
0 votes
2 answers
147 views

Evaluate the following improper integral $\int_0^\infty (\ln(1-e^{-x}))^ce^{-ax}x^bdx, \quad a>0, b,c \in \mathbb{N}_0$

Evaluate the improper integral $$ \int_{0}^{\infty}\ln^{\large c}\left(1 - {\rm e}^{-x}\right) {\rm e}^{-ax}x^{b}\,{\rm d}x, \quad a > 0,\ b,c \in \mathbb{N}_{0} $$ Is it possible to find closed ...
minimax's user avatar
  • 45
0 votes
1 answer
103 views

Prove this recurrence relation for $I_n=\int_0^\infty\frac{\operatorname du}{(1+u^2)^n}$ [duplicate]

Problem. Define $I_n=\int_0^\infty\frac{\operatorname du}{(1+u^2)^n}$. Prove that $(2n-1)I_n=2nI_{n+1}$. I tried to integrate by parts two times (the second time in the "$\cdots$") \begin{...
杜豆豆's user avatar
1 vote
0 answers
26 views

Improper integral convergence with parametric bound

I have an exercise asking for the convergence of an integral with a parametric bound $\alpha$: $$ \int_0^\alpha \frac{e^{-x}}{x-\sqrt{x}-2} dx$$ The domain of the function I get: $$ x - 2 \neq \sqrt{x}...
Michele Scattola's user avatar
0 votes
0 answers
35 views

Are there examples of convergent improper integral but we can not apply FTC?

I want to find examples of improper integral such that (1) the lower and upper limits of the integral are finite (2) the improper integral is convergent, (3) the integrand is expressed as one equation,...
qkqh's user avatar
  • 101
1 vote
1 answer
86 views

For which $f$ do we have $\int_{−∞}^{+∞} f (x + m) \, \mathrm{d}x = \int_{−∞}^{+∞} f (x) \, \mathrm{d}x$?

So, I want to believe that there is a theorem like so: $$∀ m. \int_{−∞}^{+∞} f (x + m) \, \mathrm{d}x = \int_{−∞}^{+∞} f (x) \, \mathrm{d}x$$ proof by infinity Assume $∀ x. ∞ + x = ∞$. $$\int_{−∞}^{+∞}...
Ignat Insarov's user avatar
0 votes
0 answers
76 views

How can I evaluate $ \int_0^\infty \frac{\ln(2e^x-1)}{e^x-1} dx $ using Feynman's trick? [duplicate]

I would like to find out a solution using just Feynman's method and Calculus I basic methods, to share with a novice student. I have found at Quora and this site some solutions containing ...
Arthur's user avatar
  • 17
-1 votes
1 answer
309 views

Evaluating: $ \int_0^{\infty}\left[1+\frac{\alpha}{\kappa}\left(\sqrt{p^2+1}-1\right)\right]^{-\kappa-1} p^2 \,dp $

I try to evaluate the following integral : $$ \operatorname{I}\left(\kappa,\alpha\right) = \int_{0}^{\infty} \left[1 + \frac{\alpha}{\kappa}\left(\sqrt{\,{p^{2} + 1}\,} - 1\right)\right]^{-\kappa - 1}\...
Gallagher's user avatar
  • 233
2 votes
3 answers
72 views

Decomposing a Fraction Involving Cube Roots for Integration

I had an exam the other day and there was this question to decide whether the following function is improperly integrable from 0 to 1. I wrote a solution for it but now I came to understand it's not ...
Shayan Koohi's user avatar
5 votes
1 answer
90 views

Show that $\int_0^1\frac{1}{(x+1)(x+2)\sqrt {x(1-x)}}$ is convergent.

Show that $\int_0^1\frac{1}{(x+1)(x+2)\sqrt {x(1-x)}} \,dx$ is convergent. The points $0$ and $1$ are the only points of infinite discontinuities of $\frac{1}{(x+1)(x+2)\sqrt {x(1-x)}}.$ The integral ...
Thomas Finley's user avatar
0 votes
0 answers
45 views

Integrals of asymptotic functions

If $f\sim g$ then these two integrals $$\int_a^{\infty}f(x)dx\text{ and } \int_a^{\infty}g(x)dx$$ are either both convergent or both divergent. Is this theorem correct? I have doubts because I think I ...
PNT's user avatar
  • 4,196
5 votes
2 answers
188 views

How to solve the integral $\int_0^\infty\frac{e^{-x} \sin x}{(e^{3 x} + 1) x^{3/10}} dx$

$$ \mbox{How to solve the following integral ?}:\quad \int_{0}^{\infty}\frac{{\rm e}^{-x}\sin\left(x\right)}{\left({\rm e}^{3 x} + 1\right)x^{3/10}}{\rm d}x $$ I think it cannot be solved using ...
stephan's user avatar
  • 375
1 vote
0 answers
22 views

Prove that the convolution of the signals and its time reversal is an odd signal.

Suppose signal $g(t)$ is obtained by time reversal of signal $f(t)$ for all times $t$. Prove that the convolution of the signals $f$ and $g$ is an odd signal. My attempt at proof Given: $g(t)=f(-t)\...
Awe Kumar Jha's user avatar
3 votes
1 answer
66 views

Compute the integral: $\int_{0}^{\infty}e^{-x}J_{0}(x)dx$ [duplicate]

At first, I was computing the following integral: $$\int_{0}^{\infty}e^{-x}J_0(\sqrt{x})dx$$ which can be easily solved using the taylor series of $J_0(x)$: $$J_0(x)=\sum_{n=0}^{\infty}\frac{(-1)^nx^{...
Silver's user avatar
  • 861
4 votes
3 answers
95 views

Compute the integral: $\int_{0}^{\infty}e^{-x}\text{Ei}(-x)dx$

My idea was to use the Laplace transform using this identity: $$\mathcal{L}\left\{\int_{0}^{t}f(x)dx\right\}=\frac{\mathcal{L}\{f(t)\}}{s}=\frac{F(s)}{s}$$ My first approach was utilizing the proof of ...
Silver's user avatar
  • 861
4 votes
1 answer
204 views

Integration $\int_0^\infty \frac{x^{a-1}}{1-x} dx$.

I tried to compute the following integral: $$ \mbox{For}\quad 0 < a < 1,\qquad \int_{0}^{\infty}\frac{x^{a - 1}}{1 - x}{\rm d}x $$ $$ \mbox{We know that}\quad \int_{0}^{\infty}\frac{x^{a - 1}}{1 ...
wwwwww's user avatar
  • 81
2 votes
1 answer
76 views

Integral involving product of erfc and exponential

I want to calculate this integral: $$I = -k\int_0^\infty \text{erfc}\left({\frac{\eta + z + 2akt}{2\sqrt{at}}}\right)\exp\left[k(\eta + z + akt)\right]\ d\eta$$ In his Handbook of Linear Partial ...
HiddenBabel's user avatar
1 vote
2 answers
116 views

Any method to evaluate or prove the divergence of $\int_0^{\infty} \frac{x^n \tan ^{-1} x}{1+x^2+x^4} d x$?

Recently, I met a difficult integral in the post $$ \int_0^{\infty} \frac{x^2 \tan ^{-1} x}{1+x^2+x^4} d x = \frac{\pi^2}{8 \sqrt{3}}+\frac{\pi}{24} \ln \left(\frac{2-\sqrt{3}}{2+\sqrt{2}}\right)+\...
Lai's user avatar
  • 22.3k
-1 votes
2 answers
127 views

Evaluate integral $\int_0^1 \frac{\ln(1-x)+x}{x^2}~dx$ [closed]

Evaluate : $$\int_0^1 \frac{\ln(1-x)+x}{x^2}~dx$$ I had the idea of using convergence, split it in two integrals, but I can't end it.
Pics Deb's user avatar
1 vote
4 answers
130 views

Convergence of $\int_{0}^{\infty} \frac{\log\left(x\right)}{x^{2} - 1}{\rm d}x$

I'm trying to solve the following complex analysis problem: $$ \mbox{Show that}\ \forall\ n > 1\mbox{, the integral}\quad \int_{0}^{\infty}\frac{\log\left(x\...
Giovanni Petrone's user avatar
0 votes
1 answer
57 views

Rigour, Intuition for Lebesgue Integrability

I'm a second year undergraduate maths student. I have some difficulty determining the Lebesgue integrability of some functions. I think my main issue is that I don't know how to rigorously determine ...
CatsAndDogs's user avatar
0 votes
0 answers
46 views

Validity of the general solution to $\int_{-\infty}^{\infty} e^{-ax^2-bx}dx$

I was looking for a general solution to use as shortcuts, and I got the answer online but I don't know what $a$ and $b$ can be so that this is true: $$\int_{-\infty}^{\infty} e^{-ax^2-bx}dx=e^{\frac{b^...
Ivy's user avatar
  • 87
6 votes
4 answers
186 views

Find out $\int_0^\infty\frac{x^2+ax+1}{1+x^4}\arctan\frac1x\,{\rm d}x$.

$$ \mbox{Find out}\quad\int_{0}^{\infty}\frac{x^{2} + ax +1}{1 + x^{4}}\,\arctan\left(\frac{1}{x}\right){\rm d}x $$ My attempt: \begin{align}I &= \int_{0}^{\infty} \frac{x^2 + ax + 1}{1 + x^4}\,\...
Ash_Blanc's user avatar
  • 1,133
2 votes
1 answer
140 views

Proper way of computing $\int_{0}^{\infty}\frac{\ln(\cos^2(x))}{x^2}dx$??

I noticed that people solved the following integral using Lobachevsky trick. $$ \mbox{However, notice that in}\quad \int_{0}^{\infty}\frac{\sin^{2}\left(x\right)}{x^...
Silver's user avatar
  • 861
2 votes
3 answers
121 views

Cheap way of estimating $\int_0^{+\infty}\frac{\sin^4x}{x^4}\operatorname dx$

Let $I=\int_0^{+\infty}\frac{\sin^4x}{x^4}\operatorname dx$, find the integer part of $10I$. In fact, it is possible to show that $I=\frac\pi3$. But I think the process is too technical. Since the ...
youthdoo's user avatar
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