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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2
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1answer
20 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
votes
3answers
72 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
votes
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 ...
3
votes
1answer
56 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
vote
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
votes
1answer
38 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
votes
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 = ...
3
votes
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
vote
1answer
29 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
votes
1answer
110 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 ...
0
votes
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 $\int_1^\infty \left(\frac{1}{u} - \frac{1}{u}\right)du$, ...
1
vote
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
votes
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 ...
1
vote
0answers
81 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 ...
0
votes
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 ...
1
vote
1answer
44 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
vote
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
votes
2answers
30 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 ...
-1
votes
1answer
38 views

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

Can you help me to test this integral for convergence, please $$\int\limits_1^e \frac{1}{\sqrt{1 - \ln^2x}}\,dx$$
1
vote
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
vote
1answer
31 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
vote
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 $0$ ...
0
votes
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} + ...
0
votes
1answer
44 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 ...
3
votes
2answers
46 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
133 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
vote
1answer
42 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 $$ ...
1
vote
1answer
36 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 $p \in (-1, 1)$. ...
0
votes
1answer
23 views

Method used for improper integrals can be applied to proper integrals also?

If $f$ is continuous on $[a,b]$, show that $$\lim_{c\to a^+}\int_{c}^{b}f(x)dx=\int_{a}^{b}f(x)dx$$ Hint: A continuous function on a closed finite interval is bounded and there exists a ...
1
vote
1answer
31 views

Improper Integral: Comparison Test

I have the following improper integral: $$\int ^\infty _{-\infty}\frac{2016}{e^x+e^{-x}} \, dx$$ My question is how to prove that it is convergent or divergent by using the Comparison Test.
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0answers
16 views

Prove improper integral converges

I'm studying the behaviour of the Bessel function as $x \rightarrow \infty$ part of the assignment requires me to prove the following: Prove the improper integrals $$\int_x^\infty ...
-1
votes
1answer
47 views

Find the value of p for which the integral converges and evaluate integral for $\int ^\infty_e \frac{1}{x(\ln x)^p} dx$

Find the value of p for which the integral converges and evaluate integral for $\int ^\infty_e \frac{1}{x(\ln x)^p} dx$ MY ATTEMPT: Given Integral: $I= \int ^\infty_e \frac{1}{x(\ln x)^p} dx$ put ...
0
votes
1answer
20 views

Integral of reciprocal of absolute value

I am having trouble with the following question. For which values of $n, p$ is the integral $\int_{\mathbb{R}^n}\frac{1}{|x|^p}$ convergent? I need to derive a relationship between $n$ and $p$, i.e. ...
0
votes
1answer
29 views

Would the sum after applying the integral test be equal to the sum of a series?

I know that in order to apply the integral test for convergence or divergence a function $f(x)$ must be positive, continuous, and decreasing. However, I was wondering if $$ \int_{1}^{\infty}f(x)\, ...
0
votes
0answers
43 views

Calculate non-elementary integrals

I'd like to calculate $\int_{-\infty}^{\infty}\sin(x^2)dx$ and $\int_{-\infty}^{\infty}\cos(x^2)dx$ I think it may be possible to do it by using the fact that: $$\int_{\gamma}^{} e^{-z^2}dz =0$$ For ...
1
vote
3answers
170 views

Integrate $\int_0^{2 \pi} \frac{2}{\cos^{6}(x) + \sin^{6}(x)} dx$

Integrate $$\int_0^{2 \pi} \frac{2}{\cos^{6}(x) + \sin^{6}(x)} dx$$ I have tried to do Tangent substitution, but there is huge power. What method is better to use here?
0
votes
1answer
47 views

Improper integral.

I have example which asks to determine whether the improper integral converges or diverges, and if it converges I have to solve it. I couldn't solve this, but I found a very descriptive solution, but ...
3
votes
2answers
34 views

difference between two improper integrals

I can't grasp the difference between $\int_{-\infty}^{\infty}\,f(t)\,dt$ and $\lim\limits_{x \to \infty} \int_{-x}^x\,f(t)\,dt$ for example if $ f(t)=t $ then the first one will give a divergent ...
1
vote
4answers
96 views

How to prove $\int_{0}^{\infty}xe^{-x}dx$ converges without calculating the limit

I am given $\int_{0}^{\infty}xe^{-x}dx$ and asked to prove that it converges and if it does, calculate the integral. I have calculated the integral and it gives 1. However, I cannot find a way to ...
5
votes
0answers
88 views

Complicated exponential integral

I encountered the following integral, which I am unable to solve: $$\int_0^\infty {x\over a} e^{{x \over a}} e^{{-b \over a}(e^{{x \over b}}-1)} e^{{-c \over d}(e^{{x \over c}}-1)} \mathrm{d}x$$ ...
2
votes
1answer
103 views

Convergence of $\int\limits_1^{\infty}x^\alpha\sin(x^\beta)dx$

I need to show that $$ \int\limits_1^{\infty}x^\alpha\sin(x^\beta)dx $$ converges if and only if $\alpha + 1 < \beta$. I used substitution and integration to get $$ ...
0
votes
1answer
84 views

Can $\int_0^1\frac{1}{t}e^{-t} dt$ be analytically or numerically integrated?

The following integral has a singularity at $t = 0$ as in this situation the exponential term becomes $1$ and it no longer dominates the $\frac{1}{t}$ term. $$f(x) = \int_0^1\frac{1}{t}e^{-t}dt$$ So ...
2
votes
0answers
108 views

Calculation of a double integral [closed]

Can you help me to calculate this integral. $$\begin{align}I&=\int_{0}^{1}\frac{1}{x^2+1}\left(\int_{0}^{1}\frac{dy}{1+xy}\right)dx\\ &=\int_{0}^{1}\frac{ln(x+1)}{x(x^2+1)}dx\\ ...
2
votes
2answers
60 views

How to compute this integral without the use of the error function?

I was watching this: https://youtu.be/qQ-56b_LvOw?t=4484 And this integral came up. $$\int_{0}^{\infty}{(2x^2+1)e^{-x^2}}dx$$ To which the answer was $\sqrt{\pi}$. They made it clear that you ...
0
votes
1answer
46 views

How to show this sequence is a delta sequence?

Consider the sequence $(\phi_n)_{n\in \mathbb{N}}$ of test-functions $\phi_n\in \mathcal{D}(\mathbb{R})$ defined by $$\phi_n(x) = \dfrac{n}{\sqrt{\pi}}e^{-n^2x^2}.$$ I want to show that this is a ...
1
vote
3answers
128 views

Computing an improper integral involving polynomial multiplied by an exponential

in my calculus class we are currently dealing with improper integrals and I was tackled with the following: $$ \int_{0}^{\infty} x^4 e^{-x^2}dx = $$ and $$ \int_{0}^{\infty} x^5 e^{-x^2}dx = $$ I ...
1
vote
3answers
49 views

Difficult integral of exponential function

Is there a closed-form solution for integrals of the form $$ V(a,b,c) := \int_a^\infty \exp( - b \, x^c) \, \mathrm{d}x \quad a,b,c > 0 . $$ Only for the special case $c=1$ I can figure out ...
0
votes
1answer
40 views

How to compute this integral with contour integration?

Consider the function $$g(z)=\dfrac{e^{izt}\phi(z)}{z},$$ where $\phi$ is a $C^\infty$ function. I want to compute the integral $$I=\int_{-\infty}^{\infty}\dfrac{e^{ixt}\phi(x)}{x}dx,$$ where $t$ ...