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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3
votes
3answers
153 views

Improper integral from 1 to infinity==>integrated function converges towards zero?

Let $f: [1, \infty) \to \mathbb{R}$ be a continuous function such that the improper integral $$\int_1^\infty f(x) \ dx$$ exists. Show or disprove that $\lim \limits _{x \to \infty} f(x) =0$. Our ...
1
vote
0answers
24 views

limit of an improper integral by comparison theorem

I am studying an integral using comparison text. I have managed to show it easily that for $b < 1$, $$\int_0^1 \frac{ln(1+x)}{x^b}dx$$ convergens and this is because I am well aware of functions ...
1
vote
1answer
53 views

Can anyone help with these improper integrals? [on hold]

$$\iiint_D{{e^{\sin(x+y+z)} \over x^2+y^2+z^2}dxdydz} ;D=\{(x,y,z),x^2+y^2+z^2\geq 1\}$$ For this one and the next it seems like spherical coordinates should be applied and in this case the $D_n={1 ...
0
votes
0answers
16 views

Determining convergence of integral with $\ln{x}$

Consider $\int_1^2 \frac{ln(x)}{(x-1)^P} dx$. I want to show that this converges only if $P \in (0,2)$. I have managed to do so for $P \le 1$, as I compare it to functions that are similar to it, ...
0
votes
2answers
27 views

Show that $\int_0^{a} e^{1/x}x^p dx $ diverges for all $p$

I'm trying to solve this problem using the convergence test: noting that for $a \ge 1 $ we have $0 \le e^{1/x} \le e^{1/x}x^p$ on the interval $[0,a]$ and so $\int_0^{a} e^{1/x} dx \le \int_0^{a} ...
1
vote
1answer
46 views

Does $\int_1^2 \frac{\ln(x)}{x-1} dx$ converge and what test is used?

$$\int_1^2 \frac{\ln(x)}{x-1} dx$$ How does one determine convergence of this? I am not interested in the value of it. I tried comparing to $1/(x-1)$ but the integral related to that diverges, and I ...
0
votes
0answers
13 views

Computing Impropoer Integral Of Gamma Distribution

I am trying to compute $$ \begin{align*} \mathrm{E}[X^2] &= \lim_{t\to\infty} \int_{0}^{t} x^2 \frac{\lambda^rx^{r-1}\exp(-\lambda x)}{\Gamma (r)}dx \\[2em] &= \frac{\lambda^r}{\Gamma (r)} ...
3
votes
0answers
45 views

Looking for a rigorous treatment of improper multiple Riemann integrals

I'm studying undergraduate-level differential and integral calculus and have recently come across the topic of improper Riemann integrals. I'm familiar with the concept for single-variable functions, ...
2
votes
1answer
29 views

equality between variable and integral

I received the following question as part of my homework: Let $f(x)$ be a continuous function onto $[0,1]$. $f(x)\le\frac{1} {2\sqrt{x}}$ for every $0<x\le1$. Prove that x=0 is the only solution ...
4
votes
1answer
70 views

Solve integral with exponent

How to solve integral: $$\int^\infty_0e^{-\frac{At^2}{t+1}}~dt , \quad A>0$$
2
votes
0answers
31 views

Are Riemann-integrals supposed to have a finite value?

Specifically, I am dealing with a task that implies that constant functions are not Riemann-integrable over $[0, \infty[$ unless $f=0$. Is that true? I didn't manage to find anything on that ...
2
votes
2answers
36 views

Convergence of improper integral $\int_{0}^{1}\frac{\log(x)}{1-x^2}dx$

Find whether the integral converges or diverges. $$\int_{0}^{1}\frac{\log(x)}{1-x^2}dx$$ I simplified it to $$\int_{0}^{1}\frac{\log(x)}{(1-x)(1+x)}dx$$ Here I have $2$ "bad" bounds (both $0$ and ...
0
votes
1answer
27 views

The maximum value (peak) of multiple self-convolution of rectangular function

In Multiple self-convolution of rectangular function - integral evaluation, formula for self-rectangular function of rectangular function seems to have been derived. How do we prove that this formula ...
-1
votes
1answer
48 views

Check whether the integrand is continuous when evaluating improper integrals

In order to evaluate improper integrals, I need to know whether the integrand is continuous between the limits of the integral. For the lower and upper limits, I believe you find out if it's ...
2
votes
1answer
34 views

Nature of an improper integral

I want to study the convergence of this integral at 0: $$ \int_0^{1}\frac{e^{\frac 1 t}}{\sqrt{t(1+t^2)}}\;dt. $$
3
votes
0answers
28 views

Integral representation of Bessel function K

Does someone have an idea how to connect the following function (appearing in the quantization of a real scalar field in a uniformely accelerated frame) : $$ K(x,y) = \int_{0}^{\infty} \frac{dt}{t} ...
0
votes
0answers
45 views

Applying contour integration to $\int_{0}^{\pi}dx\frac{cos(x)}{\sqrt{x^2 + x_0^2}}$

Is it possible to apply contour integration to find the value of following integral $$\int_{0}^{\pi}dx\frac{cos(x)}{\sqrt{x^2 + x_0^2}}$$
4
votes
0answers
121 views

How to compute this triple integral?

I am trying to do this triple integral $$\int_{0}^{\infty }\int_{0}^{\infty }\int_{0}^{\infty }(u+w)e^{-\frac{(u+w)^2}{2}}(v+w)e^{-\frac{(v+w)^2}{2}}(u+v)e^{-\frac{(u+v)^2}{2}}e^{-(\mu +\lambda ...
1
vote
0answers
41 views

Integral of $\frac{\exp\left(\, -\alpha x\,\right)\, (x-x_0)} {{(x-x_0)^2+\beta^2}}$ [closed]

Does the following integral have a closed form solution? $$ \int_{0}^{\infty} \frac{\exp\left(\, -\alpha x\,\right)\, (x-x_0)} {{(x-x_0)^2+\beta^2}}{\rm d}x $$ where $\alpha$, $\beta$ and $x_0$ are ...
2
votes
1answer
29 views

Find the inverse fourier transform of simple function

Suppose that the fourier transform of a signal $x(t)$ is $\hat x(u)=\frac{1}{2u_m}(1+\cos (\frac{\pi u}{u_m}))$ where $u_m \geq |u|$.$t$ here stands for time so $t \geq 0$ We sample the original ...
0
votes
2answers
20 views

Improper integrals using comparison theorem

In their working out I understand for the numerator 2+cosx=3 as cosx is less than equal to 1 but in the denominator I don't understand how they got from 3 square root x-x squared sinx all the way ...
2
votes
2answers
65 views

What is the largest function whose integral still converges?

Let C be the set of all functions $f(x)$ whose integral converges, i.e. for some constant $x_0$: $$\int_{x_0}^\infty f(x) dx < \infty$$ While playing with integrals in Wolfram Alpha, I noticed ...
0
votes
0answers
25 views

Dirichlet integral using real Analysis

The teacher made this approach to solve the Dirichlet integral , $$ J_n= \int_0^\frac{\pi}{2} \frac{\sin(2nx)}{\sin x}\:\mathrm{d}x,\quad I_n = \int_0^\frac{\pi}{2} \frac{\sin(2n+1)x}{\sin ...
2
votes
3answers
77 views

Compute the integral $\int_{0}^{\infty} \frac{(1 + x + x^2)}{(1+x^4)} dx $ with a residue on suitable contour.

I believe that I could try to compute the same integral with limits from $-\infty$ to $\infty$ using residue on a half circle and then let the radius tend off to infinity, and once I have that value I ...
1
vote
1answer
66 views

Does the improper integral $\int_{0}^{\infty}\sin(x^2)\;\mathrm dx$ converge? [duplicate]

Does the improper integral $\int_{0}^{\infty}\sin(x^2)\;\mathrm dx$ converge? So if it converges then $\lim_{b \to\infty}\int_{0}^{b}\sin(x^2)\;\mathrm dx$ exists and our integral converges to this ...
0
votes
1answer
36 views

Calculating Fourier Transform of $\sum_{n=1}^{3}\sin(2\pi \frac{n}{8}\frac{t}{T})$

This question deals with finding the Nyquist Frequency of a given signal. Suppose you have the signal $x(t)=\sum_{n=1}^{3}\sin(2\pi \frac{n}{8}\frac{t}{T})$ in the time domain where $T>0$ is some ...
-1
votes
0answers
18 views

How to evaluate the area of the curve?

I want to evaluate the area of the following between $x=0$ and $x=1$: $$f(x)=\frac{\log(1+x)}{1+x^2}$$ I started by introducing a suitable trigonometric substitution but according to what I did I ...
1
vote
3answers
81 views

Evaluating an integral using Gamma function [closed]

For $r \in (0,2)$, I would like to evaluate the integral $$\frac{2}{r} \int_0^{\infty} \frac{\sin(u)}{u^r} du.$$ The answer should be $$\frac{\pi \cdot \mathrm{cosec}{\frac{r\pi}{2}} ...
0
votes
0answers
9 views

if $F(s_{0})$ for some $s_{o}$exists then it exists for all $s>s_{o}$

if laplace transform $F(s_{0})$ for some $s_{o}$exists then it exists for all $s>s_{o}$. i need to prove this . now, ...
1
vote
1answer
27 views

Conditions on $f(t)$ so that $\int_{-\infty}^\infty f(t) \operatorname{sinc}(t-a) \operatorname{sinc}(t-b) dt$ converges.

Let us consider $$\int_{-\infty}^\infty f(t) \operatorname{sinc}(t-a) \operatorname{sinc}(t-b) dt \ \ \ \ (*)$$ for $a,b\in \mathbb R$. If $f\in L^1(-\infty,\infty)$ the integral converges: ...
2
votes
3answers
47 views

How to prove and evaluate an Improper Integral

How to show that this improper integral converges and how to compute its value? $$ I=\int_{0}^{\frac\pi 2}\frac{\cos(2t)}{\sqrt{\sin(2t)}}\mathrm{d}t. $$ I used that the integrated function is odd so ...
0
votes
4answers
63 views

Evaluate $\lim _{x\to \infty }\frac{1}{x}\int _0^x\cos\left(t\right)dt\:$

I think that $\lim\limits_{x\to \infty }\frac{1}{x}\int _0^x\cos\left(t\right)dt\:$ is divergent, I can prove with taylor series?
26
votes
2answers
446 views

On the inequality $ \int_{-\infty}^{+\infty}\frac{(p'(x))^2}{(p'(x))^2+(p(x))^2}\,dx \le n^{3/2}\pi.$

$ p(x)\in\mathbb{R[X]} $ is a polynomial of degree $n$ with no real roots. Show that: $$\int\limits_{-\infty}^{+\infty}\dfrac{(p'(x))^2}{(p'(x))^2+(p(x))^2}\,dx \leq n^{3/2}\pi.$$ It's easy to ...
4
votes
3answers
123 views

Compute $\int_{0}^{\infty}\frac{x \log(x)}{(1+x^2)^2}dx$

Given $$\int_{0}^{\infty}\frac{x \log(x)}{(1+x^2)^2}dx$$ I couldn't evaluate this integral. My only idea here was evaluating this as integration by parts. \begin{align} \int\frac{x ...
3
votes
1answer
40 views

How to compute $\int _\mathbb{R}\frac{sin^{2n}t}{t^{2n}}dt$?

If $n=1$ we can compute $\int _\mathbb{R} \frac{sin^{2}t}{t^{2}}dt$ by using Parseval's formula since $\widehat{1_{[-1,1]}}(x)=2\frac{\sin x}{x}$. We obtain $\int _\mathbb{R} ...
2
votes
5answers
101 views

How we can solve that $\lim _{_{x\rightarrow \infty }}\int _0^x\:e^{t^2}dt$?

How we can solve that $\lim _{_{x\rightarrow \infty }}\int _0^x\:e^{t^2}dt$ ? P.S: This is my method as I thought: $\int _0^x\:\:e^{t^2}dt>\int _1^x\:e^tdt=e^x-e$ which is divergent, so all your ...
1
vote
1answer
89 views

Hints for evaluating $ \lim_{y \to +\infty}y \int_0^{+\infty}{e^{-x^2}\sin(2xy) dx}$ [closed]

Please give me some hints for this limit. $ \lim_{y \to +\infty} y\int_0^{+\infty}{e^{-x^2}\sin(2xy) dx}$
2
votes
2answers
69 views

equation for the beta function

Using only the definition $$B(x, y) = \int_0^1 t^{x-1}(1-t)^{y-1}dt$$ for the Beta function, proof the term: $(x + y)B(x + 1, y) = xB(x, y) \space\space \forall x, y > 0$ . Thanks in advance! ...
2
votes
1answer
30 views

Convergence and value of improper integral

Show, that the integral $\int_0^\infty e^{-x^a}dx$ exists for all $a > 0$, and show that it's value is $\frac{1}{a}\Gamma(\frac{1}{a})$ where $\Gamma(x)$ is the gamma function. I've tried ...
1
vote
1answer
23 views

convergence of a integral using comparation

analyze the comparison criterion if the integral converges or not $\int\limits_{2}^{+\infty}\frac{\cos x}{2+e^{x^3}}dx$ attempt i used the fact that $-\frac{1}{2+e^{x^3}}\le\frac{\cos ...
1
vote
0answers
40 views

Calculate an integral with delta function

In order to calculate the integral $$ f(x) = \frac{2}{\pi}\int_0^{\pi/2}\delta\Big(x-\sqrt{1\pm\sqrt{1-\beta^2\sin^2t}}\Big)\mathrm{d}t $$ where $\beta\in(0,1]$. I am hunting for a better solution, ...
-1
votes
1answer
81 views

Integral identity involving sin(x)/x

Prove or disprove $$\displaystyle\int_{-\infty}^{\infty} \frac{3 \sin \left( x\right )}{x} \mathrm{d}x = \int_{-\infty}^{\infty} \frac{4 \sin ^ 3\left( x\right )}{x^3} \mathrm{d}x$$
1
vote
1answer
26 views

existence of an improper integral

Let $f: [1, \infty ) \to \mathbb{C}$ be a continuous function with a bounded antiderivative $F(x)$ on $[1, \infty)$. Show, that the integral $$ \int_1^\infty \frac{f(x)}{x^s} dx$$ exists for each $s ...
0
votes
1answer
20 views

Leibniz rule for an improper integral

It follows from leibniz rule that if $\frac{\partial f}{\partial \theta_0}(\theta,\theta_0)$ exists then $$\frac{d}{d\theta_0}\bigg(\int_0^{\theta_0}f(\theta,\theta_0)d\theta\bigg)=\int ...
1
vote
1answer
75 views

Which of the following is true for $\int_{1}^{0} x\ln x\, \text dx$?

Which of the following is true for $\int_{1}^{0} x\ln x\,\text dx$ it is equal to $−1/4$ it is divergent it is equal to an irrational number does not have a closed form it is impossible to ...
3
votes
1answer
45 views

Improper Integral with trigonometric functions

Determine if the following integral converges: $$\int_{-\infty}^{\infty}\frac{\cos(x)}{x^3+4x}dx.$$ So far I've thought about using the comparison test but I'm not sure how to implement it. My first ...
5
votes
1answer
114 views

Improper integral Riemann sum limit in the derivation of Fourier series to Fourier transform

To give background to my question, in all the books I've looked at to derive the inverse Fourier transform of a continuous function $f$ on $\mathbb{R}$, they seem to work as follows. Let $k$ be a ...
1
vote
1answer
27 views

proof related to convergence of a integral

i have the following condition $$0\le f(x)\le g(x)$$ and $$\int_{a}^{b}g(x)dx$$ is convergent for any $a$ and $b$ (which means $a$ or $b$ can tend to infinity) then prove that $$\int_{a}^{b}f(x)dx$$ ...
5
votes
4answers
132 views

Difficult improper integral: $\int_0^\infty \frac{x^{23}}{(5x^2+7^2)^{17}}\,\mathrm{d}x$

How can I find a closed-form expression for the following improper integral in a slick way? $$\mathcal{I}= \int_0^\infty \frac{x^{23}}{(5x^2+7^2)^{17}}\,\mathrm{d}x$$
0
votes
1answer
26 views

Integration with Respect to the Floor Function

Let $[x]$ be defined as the greatest integer part of $x \in \mathbb{R}$. Let $0<t<1$ and $\alpha(x) = [1/x]$. Compute the integral: $I(t) = \displaystyle\int_{t}^{1}x^{a}\mathrm{d\alpha(x)}$ ...