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 ∞ or −∞, or as both endpoints approach limits.

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3answers
42 views

For what values of K, is the integral improper?

For what values of $K$ ($K > 0$), is the following integral improper? $$\int_{0}^{K}\frac{x}{x^2-2}$$ Now, I know that the function is undefined at $x=\sqrt{2}$. I also figured out that the ...
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1answer
27 views

How to find the inverse Fourier transfmation of $\exp(-sk)/k$.

I've tried this with the help of hint given by one of my friend.He told me to first find the Inverse fourier transformation of $\exp(-sk)$ which is $$ \frac{\sqrt2}{\sqrt \pi}\frac{x}{x^2+ s^2}$$ ...
2
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1answer
48 views

Convergence of $\int_{0}^{1} \frac{\sqrt {e^2+x^2} - e^{\cos x}}{\tan^ax}dx$

The problem I'm facing is as it follow: For which values of $a$ the integral converges: $$\int_{0}^{1} \frac{\sqrt {e^2+x^2} - e^{\cos x}}{\tan^ax}dx$$ So far I figured that if $a< 1$, the ...
1
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1answer
65 views

Evaluate for $t\in \mathbb{R}$ $\int_{-\infty}^\infty{e^{itx} \over (1+x^2)^2}dx$

Evaluate for $t\in \mathbb{R}$ $$\int_{-\infty}^\infty{e^{itx} \over (1+x^2)^2}dx.$$ Here is what I have done: Let $f(z)={e^{itz}\over (1+z^2)^2}$. This has two poles $z=i$ $z=-i$ and an essential ...
3
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2answers
51 views

Contour Integration of $\sin^2(x)/(1+x^2)$

How should I calculate this integral $$\int\limits_{-\infty}^\infty\frac{\sin^2x}{(1+x^2)}\,dx\quad?$$ I have tried forming an indented semicircle in the upper half complex plane using the residue ...
4
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4answers
130 views

Show that $\int_{-\infty}^\infty {{x^2-3x+2}\over {x^4+10x^2+9}}dx={5\pi\over 12}$

Show that $$\int_{-\infty}^\infty {{x^2-3x+2}\over {x^4+10x^2+9}}dx={5\pi\over 12}.$$ Any solutions or hints are greatly appreciated. I know I can rewrite the integral as $$\int_{-\infty}^\infty {(...
3
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1answer
46 views

Using residue theorem to integrate from $-\infty$ to $\infty$

I'm trying to integrate $$\int_{-\infty}^{\infty} {x^2 \over {(x^2 + 1)}^2(x^2 + 2x + 2)} $$ given that the function $$f(z) = {z^2 \over {(z^2 + 1)}^2(z^2+2z+2)} $$ has residues $${9i - 12 \over 100},...
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0answers
42 views

What's about $\sum_{n=1}^\infty e^{-p_n u}$, where $p_n$ is the nth-prime number?

I am assuming that the following function, for which I am asking as reference request, should be known in the literature, since Glaisher studied the Prime Zeta Function, and my computation is the ...
2
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2answers
50 views

Improper integral with module

faced with a problem when calculating the value of the integral $$ \int_{0}^{\infty} e^{-x}|\sin(x)|\, \mathrm{d}x$$ Is there any idea how?
4
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1answer
54 views

$n$-th derivative of Beta function

We pretty much know nothing about the high order derivatives of the Beta function. Well, we known for the example some recursive formulae for $\Gamma^{(n)}(1)$ as well as $\Gamma^{(n)}\left(\frac{1}{2}...
3
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4answers
99 views

Convergence of the integral $\int_0^\infty e^{-x^3} \mathop{\mathrm{d}x}$

I want to test the convergence of the integral $\int_0^\infty e^{-x^3} \mathop{\mathrm{d}x}$. There are some parts of the solution which does not make sense to me, I'm hoping that someone can explain ...
0
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1answer
28 views

Study the convergence of this improper integral

$$ \int_o^\infty t^ae^{bt}dt $$ for a,b reals. I guess I would have to separate this integral in many cases for different values of a and b. I know that if b < 0, $$ \int_o^\infty t^ae^{bt}dt &...
3
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1answer
68 views

A clean way to obtain an (analytic or numeric) solution for this integral?

A friend and I have been looking at the crazy integral $$\iiiint \limits^{\infty}_{-\infty}\exp\left[-(x-t)^2-(x-h)^2-(y+t)^2-(y-h)^2-10\right]\mathrm{d}V$$ and can't come up with a decent method on ...
5
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1answer
43 views

Can someone help me with my proof about a limit evaluation?

Problem: Let $f:[0,1[ \to \mathbb{R}$ be a non-decreasing function such that $\int_0^1{f(x)dx}<+\infty $. Show that $$ \lim_{x\to 1^-}{(1-x)f(x)}=0.$$ Proof: $f(x)$ is a monotonic function so it ...
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3answers
46 views

Prove that a function is $L^p(\mathbb{R})$

There is a specific criterion for proving that a function $f \in L^p(\mathbb{R})$ as well as proving it by definition ? Furthermore, is correct to imply that: If $|\ f|^{\ p}$ is continuous in $\...
4
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1answer
81 views

How to evaluate $\int_{0}^{\infty }\frac{e^{-x^{2}}}{\sqrt{t^{2}+x}}\mathrm{d}x$

How to evaluate the integral below $$\int_{0}^{\infty }\frac{e^{-x^{2}}}{\sqrt{t^{2}+x}}\mathrm{d}x~~~~~~(t>0)$$ The WolframAlpha gave me a horrible answer $$\frac{t}{2}e^{-\frac{t^{4}}{2}}\left \{ \...
1
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1answer
57 views

How to find all values for $\alpha$ and $\beta$ such that $\int _0^{\infty }f\left(x\right)$ converge [duplicate]

$f(x) = \begin{cases} x^{\alpha }\left(1-cos\left(1-x\right)\right)^{\beta } & \text{if $\;\;\;0<x<1$} \\ \frac{1}{x^{\alpha }+x^{\beta }} & \text{if $\;\;\;1\ge x$} \end{cases}$ I ...
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1answer
21 views

Definite Integral $\int_0^\infty\exp(-\sqrt{x^2+y^2})\left(\frac{1}{x^2+y^2}+\frac{1}{\sqrt{x^2+y^2}}\right)dx$ (mod. Bessel funs 2nd kind $K_n$?)

I'm trying to solve the definite integral $I_1=\int_0^\infty\exp(-\sqrt{x^2+y^2})\left(\frac{1}{x^2+y^2}+\frac{1}{\sqrt{x^2+y^2}}\right)dx,$ with $y>0$ and is obviously symmetric (so boundaries ...
3
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0answers
63 views

Regularizing the sum of all factorials

Consider the series $$\sum_{n=0}^\infty n! = 0! + 1! + 2! + 3! + 4! + \ldots = 1 + 1 + 2 + 6 + 24 + \ldots$$ This series clearly diverges. Now, given that the Gamma function is defined by $$n! = \...
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1answer
25 views

Is this improper integral $\int_e^\infty \frac{dt}{t^a \log^b (t)}$ convergent?

$\int_e^\infty \frac{dt}{t^a \log^b (t)} $ What I've done is that for $t > e$, $$\int_e^\infty \frac{dt}{t^a \log^b (t)} \le \int_e^\infty \frac{dt}{t^a } $$, which converges for $a > 1$. ...
2
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1answer
21 views

Improper convergence of $ cos(x)/{x^{1/2}} $

I have to evaluate the convergence of the improper integral $ \int_1^\infty \frac {cos(x)}{x^{1/2}}dx $. As the function is continuous on every $ [1, M] $, I can tell that this function is Riemann ...
3
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3answers
73 views

Does $\int _1 ^\infty\frac {f(x)} x\,dx$ converge or diverge?

Let $f(x)$ be continuous in $[1, \infty)$ and $\int_{1}^{\infty} f(x)\,dx$ converge. I need to prove or disprove this: $\int_{1}^{\infty}\frac{f(x)}{x}\,dx$ converge. I think this is true but I don't ...
0
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0answers
18 views

Numerically Integrating Singular Integrals

When using the changing of variables technique in Numerical Integration, is there a general rule/template for which substitution to use just by looking at the function. I am confused. For example if ...
0
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2answers
32 views

Convergence Of an Integral.

While finding the Fourier Transform of the unit step function $u(t)$ , I came across the following integral: $$\int_{0}^{\infty}e^{-i\omega t}dt = \left[-\frac{e^{-i\omega t}}{i \omega}\right]_{0}^{\...
3
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1answer
67 views

Convergence of Riemann sums for improper integrals

I was considering whether or not the limit of Riemann sums converges to the value of an improper integral on a bounded interval. This appears to be true in some cases when the sum avoids points where ...
1
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1answer
21 views

Double integral of the following Exponential

I am interested in the following integral $$\int_{-\infty}^{\infty}\mathrm{d}v_1\int_{-\infty}^{\infty}\mathop{\mathrm{d}v_2}v_1v_2\exp\left(-\frac{(v_1-v_2)^2}{2a}\right).$$ Does any one have any ...
0
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1answer
40 views

Beta function. What is wrong? Misunderstanding.

I am only interested in real numbers, so here $x,y>0$ real numbers. The Beta function is defined as $$B(x,y)= \int_0^1 w^{x-1}(1-w)^{y-1}dw.$$ The above integral is defined for every $x,y>0$. ...
2
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1answer
42 views

Show that $\int_{0}^{\infty} \int_{0}^{\infty} e^{-s(x+y)}f(x)g(y) \ dx \ dy = \int_{0}^{\infty} \int_{0}^{t} e^{-st}f(t-u)g(u) \ du \ dt$.

While learning how to compute the product of two Laplace transform and the inverse transform, I faced with this equality : $$\int_{0}^{\infty} \int_{0}^{\infty} e^{-s(x+y)}f(x)g(y) \ dx \ dy = \int_{0}...
3
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1answer
40 views

Integral of Exponential raised by exponential

I am interested in the following integral $$\int_{-\infty}^{\infty}\mathrm{d}x\left(1-\exp\left[\frac{-b}{\sqrt{2\pi}a}\exp\left(-\frac{x^2}{2a^2}\right)\right]\right).$$ Does anyone know how to ...
1
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1answer
31 views

Convergence of an improper integral with a parameter

I am to test the convergence of the improper integral $$ \int_{0+}^{1-} \frac{\ln(x)}{(1-x)^a} dx$$ with the parameter $a \in R$. I have some trouble doing this so I'd appreciate a full explanation so ...
2
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1answer
111 views

Prove that $\lim_{\ r\ \to \ \infty} \dfrac{r! r^x}{x(x+1)(x+2) \dots (x+r)} = \int_{0}^{\infty} t^{x-1} e^{-t} dt $

From Havil & Dyson, "Gamma: Exploring Euler's Constant", section 6.1 I can't prove the following Euler's theorem : ... on 13 October 1729, Euler had already proposed to Goldbach the ...
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0answers
37 views

What is $\int_1^\infty du/u - \int_1^\infty du/u$?

I thought that $\int_1^\infty \frac{du}{u}- \int_1^\infty \frac{du}{u} = 0$ as the two integrals are the same or because we can write it as $\int_1^\infty \left(\frac{1}{u} - \frac{1}{u}\right)du$, ...
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1answer
34 views

Is this function bounded? (Order of explosion of a function)

I have the following function: $f:[0,1]\rightarrow \mathbb{R}$ defined as $$f(x)=\int_x^1 \frac{(y-x)^{-\alpha}}{y}dy, \quad x\in [0,1],$$ where $\alpha\in (0,1/2)$ is some fixed parameter. To me it ...
0
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1answer
25 views

Find the values of $\alpha$ for which the improper integral $\iint_Df_\alpha(x,y)dxdy$ exists

Study the function $$f_\alpha(x,y)=\frac{x^\alpha}{x^2+y^2},$$ with $\alpha \geq 0$ and the domain $D\subset \mathbb{R}^2$ that in polar coordinates is given by $0\leq r \leq 1$ and $0 \leq \theta \...
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0answers
87 views

Question on an April Fool on Fourier transform

I have a question on this answer : Let $f(x) = 1$. It's easy to see that its Fourier transform is $0$ almost everywhere, so $\hat {\hat f}(x) = 0$. By the inversion theorem, $1 = 0$. I think ...
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1answer
38 views

Improper integral involving sinc function and Pochhammer symbol

Can anyone please advise me how to integrate expressions of the form $\text{sinc}\,(x) / (1-x)_n$ along the real axis? Using a CAS, one could suggest that $$ n! \int_{-\infty}^\infty \frac{\sin \pi x}...
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2answers
72 views

If $f$ is continuous with $ \int_0^{\infty}f(t)\,dt<\infty$ then which are correct?

Let $f:[0,\infty)\to [0,\infty)$ be a continuous function such that $\displaystyle \int_0^{\infty}f(t)\,dt<\infty$. Which of the following statements are true ? (A) The sequence $\{f(n)\}$ is ...
1
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2answers
58 views

How can I show that this integral converges?

So here it is: $\int\limits_2^{+\infty}\left(\cos\frac{2}{x}-1\right)dx$. I've tried to use Cauchy, Dirichlet and Abel's tests, but can't seem to figure this out. Mathematica says in converges, but ...
0
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2answers
35 views

Improper integral proof: limit of integral exists when the integral is continuous?

We're trying to prove the integral $$\int_0^1\frac{\cos x}{x^\frac12}\,dx$$ exists as an improper integral. My teacher says that in order to prove there exists the limit of $\int_a^1\frac{\cos x}{x^\...
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1answer
41 views

Test improper integral with $\ln$ for convergence [closed]

Can you help me to test this integral for convergence, please $$\int\limits_1^e \frac{1}{\sqrt{1 - \ln^2x}}\,dx$$
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1answer
31 views

Confusion when finding convergences using divergence and integral test?

I am having a bit of confusion doing the divergence and integral tests, specifically when I am trying to visualize the functions to get a better idea of why the methods work. For example, take the two ...
1
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1answer
33 views

integrating f(x)=1/x from -a to a. convergent or divergent?

we are discussing improper integrals in Calc II, and I am failing to understand why the integral from $-a$ to $a$ of $f(x)=1/x$ is not zero. Since the function is odd and thus symmetric about the ...
1
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3answers
31 views

test for conditional and absolute convergence

find value of a parameter $\alpha$ at which integral converges absolutely and at which conditional $$\int\limits_0^\infty \frac{x + 1}{x ^ {\alpha}}\sin(x)\,dx$$ We can consider 2 cases: area of $0$ ...
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3answers
58 views

An improper integral. Prove an inequation.

$$0 < \int\limits_0^\infty \frac{x^{20} + 1}{x^{40} + 1}\,dx - \frac{20}{19} < 0.05$$ I tried to use that $$ \frac{x^{20}}{x^{40}} < \frac{x^{20} + 1}{x^{40} + 1} < \frac{x^{20} + 1}{x^{...
0
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1answer
45 views

Conditional convergence $\int_{-\infty}^{+\infty} \frac{(x-1)\sin 2x}{x^2-4x+5}dx$

Explore conditional convergence $$\int_{-\infty}^{+\infty} \frac{(x-1)\sin 2x}{x^2-4x+5}dx$$ I tried $$\int_{-\infty}^{+\infty} \frac{(x-1)\sin 2x}{x^2-4x+5}dx = \int_{-\infty}^{+\infty} \frac{\sin 2x}...
3
votes
2answers
48 views

A sufficient condition for the existence of an improper integral (or a counterexample for it)

Let me try to explain the spirit of the question. The functions $f(x)=1/x^{p}$ for $0<p<1$ and $f(x)=\ln x$ have the following properties: they are in some respect 'nice' on $\mathbb{R}^{+}$, ...
2
votes
2answers
138 views

What is $\displaystyle\int_{2}^{2}\frac{dx}{x-2}$?

Evaluate the integral: $$\displaystyle\int_{2}^{2}\frac{dx}{x-2}.$$ 1)When does $\displaystyle\int_a^a f(x)dx=0$? Always? 2)Does $\displaystyle\int_a^a$ means area between $(a,a)=\emptyset$? 3) Do ...
1
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1answer
112 views

Surface Integral over a rhombus

Evaluate the integral $$\int\int_{R}(x-y)^2 cos^2(x+y)dxdy$$ where $R$ is the rhombus with successive vertices as $(\pi,0), (2\pi,\pi), (\pi,2\pi), (0,\pi).$ My attempt- I tried doing this surface ...
2
votes
3answers
43 views

Prove $\int\limits_1^\infty x^a\sin x \, dx$ diverges for $a>1$

Let $a>1$. I need to show that $$ \int_1^\infty x^a\sin x \, dx $$ diverges. I am not sure, but this is my progress We will look first at intervals $[2m\pi,(2m+2)\pi]$. Then $$ \int_{2m\pi}^{(2m+2)...
1
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1answer
37 views

Integrating the bivariate normal distribution

Let $X$ and $Y$ have the bivariate normal density function, $$ f(x, y) = \frac{1}{2 \pi \sqrt{1 - p^2}} \exp \left\{ - \frac{1}{2(1 - p^2)} (x^2 - 2pxy + y^2) \right\} $$ for fixed $p \in (-1, 1)$. ...