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.

learn more… | top users | synonyms

-1
votes
1answer
24 views

Improper integral: converges or diverges

$I= \int^{666}_{16} \frac{1}{x^{1/4} - 2} dx $ hint: ($8x^{1/4} < x$ when $16 < x$) Determine if the improper integral converges or diverges. Be sure to treat the improper integral with ...
5
votes
2answers
52 views

Is $\int_1^{\infty}\frac{x \cos(x)^2}{1+x^3}$ convergent or divergent?

For the integral $$I= \int_1^{\infty}\frac{x \cos^2(x)}{1+x^3},$$ how do I test this for convergence or divergence? I know that this an improper integral- however it cannot be solved so would need to ...
1
vote
2answers
31 views

Proving the convergence of the improper integral $\int_0^1 \operatorname{ln}(\operatorname{sin}x)dx$

I'm trying to prove that \begin{equation*} \int_0^1 \operatorname{ln}(\operatorname{sin}x)dx \end{equation*} converges. I tried to show this by decomposing \begin{equation*} ...
1
vote
0answers
9 views

Improper integral-limiting result

If $\int_{a+}^bfdx$ exists, then show that $\int_{a+}^cfdx$ exists for any $c\in(a,b)$ and $\lim_{c\to a+}\int_{a+}^cfdx=0$. I proved this as follows. (i)By Cauchy's criterion, for given ...
0
votes
3answers
55 views

Does this integral converge or diverge?

I have the $$\int_{16}^{500} \frac{1}{x^{0.25} - 2} dx,$$ and am trying to find whether it converges or diverges. I have sketched the graph and noticed that their is an asymptote at $x=16$ (hence ...
0
votes
3answers
48 views

Improper integral: converge or diverge [on hold]

$I = \displaystyle \int_{1}^\infty \dfrac{x\cos^2(x) dx}{1+x^3}$ Determine if the improper integral converges or diverges. Be sure to treat the improper integral with appropriate mathematical ...
2
votes
1answer
34 views

Closed-form expression for $\int_{0}^{1}e^{-ax(1 - bx )}x^{\alpha-1}(1-x)^{\beta - 1}dx$?

As per the title, I am looking for a closed-form expression for the integral $$\frac{1}{B(\alpha,\beta)}\int_{0}^{1}e^{-ax(1 - bx )}x^{\alpha-1}(1-x)^{\beta - 1}dx$$ where $a,\alpha,\beta>0$ and ...
0
votes
1answer
30 views

Integrating indefinite and improper integrals

I am given the integral $$\int_0^\infty\frac{\sin^4(x)}{x^2}dx$$ And I must compute it. I know that the answer is $\frac{\pi}{4}$, but I don't really know how to begin solving this. I am thinking of ...
0
votes
0answers
27 views

Solve this integral or at least find an upper bound?

Let $r,t>0$ be fixed. Let $a,b,c$ be numbers such that the following integral converges (I think $a,b,c>-1$ is OK). Then I would like to compute the following integral explicitly if possible or ...
2
votes
0answers
46 views

How to find this integral from Gaussian integral? [duplicate]

How to find $$\int_0^\infty e^{-ax^2-\frac{b}{x^2}}dx$$ using gaussian integral? I tried complete the square: $$-ax^2-\frac{b}{x^2}=-\left(\sqrt{a}x+\frac{\sqrt{b}}{x}\right)^2+2\sqrt{ab}$$, but what ...
3
votes
3answers
328 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
29 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
54 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
17 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
28 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
3answers
59 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
56 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
30 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
72 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
37 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
29 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
35 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
46 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
79 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
48 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?
27
votes
2answers
484 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
124 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
31 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
24 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 ...