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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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14 views

For which values of p does the integral converge

$\int_5^\infty \frac{dx}{x^pln|x|} $ For which values of $p$, does the integral converge? Any help?
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2answers
41 views

Can a sum of a convergent and divergent integral be convergent?

I have this problem. True or False: If $\int_a^b \big(f(x)+g(x)\big)dx$ converges, then $\int_a^b f(x)dx$ and $\int_a^b g(x)dx$ both must converge. I am pretty sure this is false, but I can't ...
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0answers
3 views

How do I calculate the integral for the point spread function of an optical system without the fresnel approximation?

For an object located at the $(x_o, y_o)$ plane, and a lens located at the $(x, y)$ plane, the image is produced at the $(x_i, y_i)$ plane. We can consider the image field $U_i$ to be the sum of the ...
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1answer
58 views

Evaluating $\int_{-\infty}^{\infty}\frac{1-a\cosh(\alpha x)}{(\cosh(\alpha x)-a)^2}\cos(\beta x)\,dx$

I would like to solve the following improper integral: $$\int_{-\infty}^{\infty}\frac{1-a\cosh(\alpha x)}{(\cosh(\alpha x)-a)^2}\cos(\beta x)\,dx$$ where $a$, $\alpha$ and $\beta$ are real constants....
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1answer
40 views

Example of function such that integral of $|x(t)|$ is $<+\infty$, and integral of $|x(t)|^2$ is $\infty$

I'm trying to look for an example of a function such that: $$ 1. \displaystyle{\int \limits_{- \infty }^{+ \infty }} \lvert x(t) \rvert \, dt < + \infty $$ $$ 2. \displaystyle{\int \limits_{- \...
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2answers
57 views

Calculate $I(t,n) = \int_{-\infty}^{\infty} \big( \frac{1}{1-jq} \big)^{n} e^{-jqt} dq$

I am trying to calculate integrals of the form: $$ I(t, n) = \int_{-\infty}^{\infty} \Big(\frac{1}{1-jq}\big)^{n} e^{-jqt} dq $$ where $j = \sqrt{-1}$. In the case when $n=1$, I have: $$ I(t, 1) =...
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2answers
39 views

Compute $I(t) = \int\limits_{0}^{\infty} e^{jx(t+j)}dx$

Hello I am struggling with the following integral: $$ I(t) = \displaystyle\int\limits_{0}^{\infty} e^{jx(t+j)}dx $$ Where $j = \sqrt{-1}$. I think the answer should be: $$ I(t) = \frac{1}{1-jt} $$ ...
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2answers
123 views

Evaluate $\int_{0}^{\pi} x\sin\big(\frac{1}{x}\big)\cos x \,dx$

I wonder if an integral of the form $$\int_{0}^{\pi} x\sin\Bigl(\frac{1}{x}\Bigr)\cos x \,dx$$ which can be further simplified to $$\int_{0}^{\pi} \frac{\sin (x^{-1})}{(x^{-1})}\,\cos x\, dx=\cos x\...
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0answers
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I'm finding it difficult solving this integral [on hold]

I've been trying to solve this integral but I'm having difficulty, any help or suggestion is welcome: $$I=\iint\limits_D f(x,y)\ dA$$ where $f(x,y)=x+2y$ and $D$ is the region bounded by $y=2x^2$ and $...
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0answers
72 views

Evaluating $\int\limits_{\mathbb{R}} x\left(1-\int\limits_{-\infty}^xe^{-t^2/2}\mathrm dt\right)^{n-1} e^{-x^2/2}\mathrm dx.$

Let $f(x) = x \left(1 - \Phi(x)\right)^{n-1} e^{-x^2/2}$, where $\Phi(x) = \int\limits_{-\infty}^xe^{-t^2/2}\mathrm dt$. I am interested in evaluating $$ \int\limits_{\mathbb{R}} f(x) \mathrm dx = \...
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1answer
23 views

Triple integral convergence

I had a calculus 3 exam recently and didn't do too well, I am trying to see how I could have approached things differently. I was asked to prove/disprove that the improper integral $\int_{-\infty}^{\...
3
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1answer
46 views

Show $\int \frac{\sin(x^p)}{x} dx = \frac{\operatorname{Si}(x)}{p} $

I was messing around with the Fresnel integral and the Sine integral and found that $\int_{0}^{\infty} \frac{\sin(x^2)}{x}dx=\frac{\pi}{4}$ but I dont see how to extend to irrational powers.
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1answer
17 views

Bounded from above integral with exponential

I am trying to bound from above the following integral $\int_{\left\{|u| \geq u_n\right\}} u^{4} e^{-\frac{u^{2}}{2n}} du$ where $u_n = 2 \sqrt{nlogn}$. Could you please give me any idea how I could ...
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1answer
78 views

Integral $\int\limits_{0}^{\infty}e^{-\frac{1}{2}\left(y^2+\frac{t^2}{y^2}\right)}\,dy$

I have no idea ho to compute this integral: $$\int\limits_{0}^{\infty}e^{\large-\frac{1}{2}\left(y^2+\frac{t^2}{y^2}\right)}\,dy$$ I have put this integral into Wolfram Mathematica and the result ...
3
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1answer
54 views

Classify $I(a,b)=\int_{0}^{\frac{\pi}{2}} \frac{dx}{\sin(x)^a\cos(x)^b}$ with $a,b\in \mathbb{R}$

Classify $I(a,b)=\int_{0}^{\frac{\pi}{2}} \frac{dx}{\sin(x)^a\cos(x)^b}$ with $a,b\in \mathbb{R}$. I'm really lost, this excercise is so different from the others and I can't find any similar ...
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4answers
134 views

Asymptotic behavior of the integral $H(\beta)=\frac{2}{\pi}\beta\int_{0}^{\infty}\exp\left(-x^{3/2}\right)\sin(\beta x)xdx$

I found the integral $H(\beta)$ (which is called Holtsmark distribution) in Holtsmark's theory of ion field in plasma. In a book there is its asymptotic representation at small and great $\beta$: $$ ...
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0answers
33 views

Studying convergence of an integral at the vary of alpha

I need to understand this exercise from my math book: study, at the vary of alpha belonging to R, the convergence of the following integral and then calculate the value for $a = 1$. $$ \int_0^{4} \...
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0answers
57 views

Cauchy principal value of $\int_{-\infty}^\infty\frac z{8-z^3} \mathrm dz$ [closed]

Let $f(z) = \frac{z}{8 - z^3}$. Then find the Cauchy principal value of $$\int_{-\infty}^{\infty} f(x)\, dx$$ Please explain it properly with basic things.
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3answers
102 views

Integral $\int_{0}^{1}\frac{\log x}{2 - x} dx$

Calculate: $$\int_{0}^{1}\frac{\log x}{2 - x} dx$$ I've done a lot of research here in the community. I tried varying variants, but I did not get anything. The problem is the $2$ present in the ...
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1answer
29 views

Non-uniform convergence of improper integrals II

I'm looking at approaches for proving or disproving uniform convergence of improper integrals. I asked this question: Techniques for proving non-uniform-convergence of improper integrals but I ...
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4answers
90 views

Problem in solving the improper integral $\int_0^\infty\frac{1}{(x+1)(ax+1)(bx+1)} dx$

I am trying to solve the following improper integral $I$. $$I = \int\limits_0^\infty\frac{1}{\left(x+1\right)\left(ax+1\right)\left(bx+1\right)} dx$$ However, I myself get the divergent answer. When ...
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1answer
52 views

Improper Integrals in Analysis

Let $f:[0,\infty) \rightarrow \mathbb{R}$ be a continuous function and let $g(x)=\frac{1}{x}\int_1^x f(t)dt$; $x>0$. Assume that $\lim_{x\rightarrow \infty} g(x)=B$ exists. Let $0 < a < b$ be ...
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3answers
38 views

A naive change of integration limit

I am trying to understand something pretty dumb. First question: Assuming I have the following integral: $$ \int_{-\infty}^{\infty}x^3e^{-\frac{-x^2}{2}}dx $$ If it was an indefinite integral I ...
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2answers
276 views

Ramanujan's Master Theorem relation to Analytic Continuation

$\DeclareMathOperator{Re}{Re}$ To provide some background, this is a question based on establishing the identity $$\int_0^\infty \frac{v^{s-1}}{1+v}\,dv=\frac{\pi}{\sin \pi s},\qquad 0<\Re s<1$$...
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2answers
63 views

For which $p,q$ does the $\int_0^{\infty} \frac{x^p}{\mid{1-x}\mid^q}dx$ exist?

For which $p,q$ does the $\int_0^{\infty} \frac{x^p}{\mid{1-x}\mid^q}dx$ exist ? Can you help me, I have been siting hours on this question . I got that for $ q<1$ and $p>q+1$, but I am not ...
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1answer
71 views

Proving $\int_0^\infty x^ne^{-tx}\frac{\sin x}xdx=\frac{\sin n\theta}{(1+t^2)^{n/2}}(n-1)!$, where $\theta=\arcsin\frac1{\sqrt{1+t^2}}$

I have stumbled upon the PDF by Leo Goldmakher from University of Toronto, Canada, named Differentiation Under The Integral Sign (PDF link). In that pdf, he gave a theorem (his Theorem 1, at the end ...
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1answer
55 views

Does $\int_{-\infty}^{\infty}f(x)\,dx$ exist?

Let $f:\Bbb R\to \Bbb C$ be a continuous map, if $\int_{-\infty}^{\infty}|f(x)|\,dx$ exists, then $\int_{-\infty}^{\infty}f(x)\,dx$ exists, doesn't it?
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1answer
66 views

An integral identity.

P. Dravek and G. Holubova, Elements of Partial Differential Equations, Section 3.4 Exercise 21 Part c): Fix $x\in \mathbb{R}$ and $y>0$. Show that: $$\int_{0}^\infty e^{-y\sqrt{1+t^2}}\cos(xt)...
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1answer
69 views

What's wrong with my approach to this improper integral?

$$\int _0^{\infty }\:\left(\frac{2x}{x^2+5}\right)-\left(\frac{6}{3x+2}\right)$$ $$\int _0^{\infty }\:\left(\frac{2x}{x^2+5}\right)-2\int _0^{\infty }\:\left(\frac{3}{3x+2}\right)$$ $$\lim _{b\to \...
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0answers
35 views

An upper bound for a multiple integral

Suppose $0 < a < b < \infty$. I am trying to find a nontrivial upper bound for: $$ \int^\infty_0 \int^\infty_0 \frac{dx \, dy}{xy + \frac{e^{a(x+y)}}{b^2}}. \tag{1} $$ Now, I know that $$ ...
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3answers
85 views

About $\int_{-\infty}^{0}\text{arcsin}\left(e^x\right)\text{d}x$

I've shown that the integral $I$ exists $$I=\int_{-\infty}^{0}\text{arcsin}\left(e^x\right)\text{d}x \approx 1.089$$ Is there a way to find the exact value ? Using $u=e^x$ we have $\text{d}u=u\text{d}...
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3answers
91 views

How to do integral $\int_{0}^{\infty} \frac{x^4 e^{x}}{(e^x-1)^2}\,dx$

$$\int_{0}^{\infty} \frac{x^4 e^{x}}{(e^x-1)^2}\,dx$$ This integral came from the Debye theory of the molecular heat capacity of crystal. I heard that this integral form is related to Riemann Zeta ...
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1answer
56 views

Compute integral in unbounded area

Compute the following integral: $$\iint_D \frac { |x-y| } { (x^2 + y^2 + 1)^2 } \, dx dy , $$ where $D = \{ (x, y) \in \mathbb R^2 \mid x \geq 0 ,\, y \geq 0 \}$. So is there an easy way to ...
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1answer
43 views

How to evaluate the integral in the following cosine Fourier transform?

Could anyone prove the following equation: $$\int^{\infty}_0 e^{-a\sqrt{x}}\cos(a\sqrt{x})\cos(bx)\,dx=a\sqrt{\frac{\pi}{8b^3}}e^{-\frac{a^2}{2b}}$$
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1answer
82 views

What is $ \int_0^{\infty} \left( e^{\frac{\ln x} {x}}-1 \right) dx$?

As per what I have tried, I don't think that $ \int\left( e^{\frac{\ln x} {x}} \right) dx$ can be expressed in terms of elementary functions, but then I again I have just started learning integration ...
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2answers
47 views

Find positive $K$ such that $\int_0^\infty\left(\frac{1}{\sqrt{2x^2+1}}-\frac{K}{x+1}\right)dx$ converges

Find positive $K$ such that $$\int_0^\infty\left(\frac{1}{\sqrt{2x^2+1}}-\frac{K}{x+1}\right)dx$$ converges I used the fact that $\frac{1}{\sqrt{2x^2+1}}-\frac{K}{x+1}>\frac{-K+1/\sqrt2}{1+x}$ ...
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2answers
38 views

A limit of sequence of improper integral

I was baffled to deal with such problem. The result of this problem is obvious but I don’t how to solve it with a precise method because too many parameters in this integral.
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2answers
24 views

$K$ integrable on $\mathbb{R}$ implies $K$ bounded?

Suppose $K:\mathbb{R} \rightarrow \mathbb{R}$ is a nonnegative function satisfying $\int_\mathbb{R}K(x)dx=1$ and $\int_\mathbb{R}K^2(x)dx<\infty$. I know that if $K:\mathbb{R} \rightarrow \mathbb{...
8
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1answer
107 views

Two Euler sums each containing the reciprocal of the central binomial coefficient

Is it possible to find closed-form expressions for the following two Euler sums containing the reciprocal of the central binomial coefficient? $$1. \sum_{n = 0}^\infty \frac{(-1)^n H_n}{(2n + 1) \...
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2answers
55 views

Studying the convergence of $\int_0^1\frac{dx}{\sqrt{1-x^4}}$

I am studying the convergence of improper integrals. In this case I am asked to calculate if the following integral converges.$$\int_0^1\frac{dx}{\sqrt{1-x^4}}$$ To do so my professor used the ...
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2answers
30 views

discuss integrability of $(x_1, \cdots, x_n) \mapsto \frac{1}{(1+x_1+x_2+ \cdots + x_n)^{\alpha}}$ on $[0,+ \infty[^n$

let's first discuss integrability of $(x,y) \mapsto \frac{1}{(1+x+y)^{\alpha}}$ on $[0,+ \infty[^2$ the integrand being positive and measurable we can compute the integral in however order we want ...
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1answer
53 views

Given $\int_0^{\infty} e^{-x^2}dx = \frac{\sqrt\pi}{2}$, find $\int_0^{\infty} e^{-\frac{x^2}{2}} dx$

Given $\int_0^{\infty} e^{-x^2}dx = \frac{\sqrt\pi}{2}$, find $\int_0^{\infty} e^{-\frac{x^2}{2}} dx$ and $\int_0^{\infty} x^2e^{-x^2}dx$ I tried to figure out a way to get the integral in the form ...
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1answer
33 views

Infinite Integral of a Product of Bessel Functions

I am interested in any analytic information about the following integral: $i^{4m+1} \int_0^{\infty} t^{1/4} J_m^4(t) J_{\nu}(\alpha t) dt$ where $i = \sqrt{-1}$ is the imaginary unit $m$ is a ...
2
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0answers
31 views

Integral involving a Gaussian and a rational function.

Let $a \ge 0$ and $b \ge 0$ be real numbers. By generalizing the approach from Evaluating $\int_{\mathbb{R}}\frac{\exp(-x^2)}{1+x^2}\,\mathrm{d}x$ . we have derived the following results. Let $n\ge ...
5
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0answers
35 views

Proof that improper integral $\int_{0}^{\infty} e^{-x}(1+\sin(x^2)) dx$ exists

I want to prove that the following integral exists: $$ \int_0^\infty e^{-x}\left(1+\sin\left(x^2\right) \right) dx $$ First I tried to calculate it by splitting it up (multiplied it out and then ...
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3answers
34 views

State whether the following improper integral converges, and, if it does, find its value.

Hello, I have been struggling with the simplest question of this topic. The integral of it is $\frac{-1}{(1+x)} dx$, and replacing infinity with the limit of $b \to \infty$, I found $\frac{-1}{(1+b)}...
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1answer
47 views

convergence of $\int_{0}^{\infty}{\frac{x^2-1}{x^2+1}\frac{\sin(2x)}{x} dx}$

How can I show that $\int_{0}^{\infty}{\frac{x^2-1}{x^2+1}\frac{\sin(2x)}{x} dx}$ converges? I tried the criteria listed in my book but none seem to apply
0
votes
1answer
36 views

Studying the convergence of $ \int_1^\infty \frac{\sin(x)}{(x-1)^\alpha}$ [closed]

How do I study the convergence of the improper integral $$ \int_1^\infty \frac{\sin(x)}{(x-1)^\alpha}$$ depending on the parameter $\alpha>0$? I have tried to separate the integral in two parts, ...
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0answers
25 views

Evaluate $\int_{-\infty}^{\infty}xe^{-x^{2}}dx$ [duplicate]

Evaluate improper integral: $\int_{-\infty}^{\infty}xe^{-x^{2}}dx$
4
votes
1answer
154 views

The integral of a function multiplied with an unbounded function converges if the product is a bounded function

Suppose $f:[1,\infty]\rightarrow \mathbb{R}$ is such that $g(x):=x^2f(x)$ is a bounded function. Prove that $\int_1^\infty f$ converges. Intuitively I think it seems reasonable. As $x^2$ is ...