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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0
votes
1answer
81 views

Integral of $\int_{-\infty}^\infty\frac{e^{-y^2/2} \sinh(cy)^2} {\cosh(Mcy) }dy$

Is it difficult to compute or find a good computable lower bound on the integral \begin{align*} \int_{-\infty}^\infty\frac{e^{-y^2/2} \sinh(cy)^2} {\cosh(Mcy) }dy \end{align*} where $c$ and $M$ are ...
4
votes
4answers
263 views

Can we determinine the convergence of $\int_0^\infty \frac{x^{2n - 1}}{(x^2 + 1)^{n + 3}}\,dx$ without evaluating it?

Can we determine convergence without evaluating this improper integral? $$\int_0^\infty {\frac{x^{2n - 1}}{{\left( x^2 + 1 \right)}^{n + 3}}\,dx}\quad\quad n\geq 1\;,\; n\in\mathbb{Z}$$ When ...
0
votes
2answers
43 views

Consider the intergal $I=\int_{1}^{\infty}e^{ax^2+bx+c}dx$, where $a,b,c$ are constants. When does the integral converge? [closed]

Consider the integral $I=\displaystyle\int_{1}^{\infty}e^{ax^2+bx+c}dx$, where $a,b,c$ are constants. When does the integral converge? As usual, these are alien concepts to me, it gets tough to ...
3
votes
1answer
56 views

Computing $\int_{0}^{\pi/2}\cos(x)\ln(\tan(x))dx$

Compute $\int_{0}^{\pi/2}\cos(x)\ln(\tan(x))dx$ It is easy to check this improper integral converges. One also notes that ...
2
votes
2answers
71 views

Manipulating $\int_{x_{0}}^{\infty} \frac{1}{x} \, \cos (x t) \, \text{e}^{-x^{2}} \, dx$

Is there a way to express the integral $I(x_{0}, t) = \int_{x_{0}}^{\infty} \frac{1}{x} \, \cos (x t) \, \text{e}^{-x^{2}} \, dx$, where $x_{0} \neq 0$ and $t \ge 0$, in terms of more well-known ...
1
vote
1answer
23 views

Proving an integral identity

I'm dealing with the Hermitian operator, and I've been asked to prove that all $f(x) = x^n e^{\alpha x}$ belong to $L^2(-\infty,\infty;e^{-x^2/2})$ by showing that: $$\int_{-\infty}^{\infty}x^m ...
0
votes
1answer
41 views

For what values is this integral convergent?

How can I find for what values of $r$ $$\int_0^\infty x^re^{-x}dx$$ converges? I started by rewriting it as $$\lim_{b\to\infty}\int_0^bx^re^{-x}dx$$ but am not sure how to figure it out from here.
0
votes
2answers
59 views

Improper integral problem.

How to find divergence/convergence condition for $p$ on $$\int\limits_{2}^{\infty} \frac{1}{{(\ln x)}^p} \, \mathrm d x$$ I tried comparison test , but failed.
0
votes
1answer
29 views

Sum of integrals with variables shifted in each sum: How to justify this expression?

I annoyingly can't justify a step in the solution of the following problem. I have the following expression at hand: $$ \sum_{n=1}^{N}\int_{-\infty}^{\infty}{(y(x_n + \xi) - t_n})\nu(\xi)\eta(x_n + ...
2
votes
3answers
88 views

How evaluate $\int \frac{\cos^2(x)}{1 + \text{e}^x}dx$ to find an improper integral

Can someone help me evaluate this: $$\int \frac{\cos^2(x)}{1 + \text{e}^x}dx\;?$$ I need it for determining whether the improper integral $\int_0^\infty {\frac{{\cos^2{{(x)}}}}{{1 + ...
4
votes
2answers
75 views

Can anyone help me with this improper integral?

$$\int_{0}^{\infty} \left(e^{-\frac{1}{x^2}}-e^{-\frac{4}{x^2}}\right) dx$$ I've tried much of the techniques used in the textbook, none have led to anything concrete, or i am not just able to see ...
1
vote
1answer
27 views

Integral divergence

I´m trying to solve this problem about integral convergence, and I would be happy for any help. I shoul find out for what values of $a$ is this integral convergent: $$\int_0^\infty ...
13
votes
2answers
212 views

Closed form $\int_{-1}^{1} \frac{\ln (\sqrt{3} x +2)}{\sqrt{1-x^{2}} (\sqrt{3} x + 2)^{n}}\ dx$

Does the following integral $$\int_{-1}^{1} \frac{\ln (\sqrt{3} x +2)}{\sqrt{1-x^{2}} (\sqrt{3} x + 2)^{n}}\ dx, \; \; n \in \mathbb{N}$$ have a nice closed form? Basically I cannot tackle it in any ...
1
vote
1answer
49 views

Divergence of improper integral $\int_{2}^{\infty} \frac{\cos(2x)\cos(6x)}{x\ln x}$

Given the integral $$\int_{2}^{\infty} \frac{\cos(2x)\cos(6x)}{x\ln x},$$ is there an easier way to show its divergence than starting to break this integral to a sum of smaller integrals using ...
0
votes
2answers
27 views

How to prove improper integral $\int_0^1 {\frac{1}{{{{(-\ln x)}^p}}}dx} $ diverges when $p>=1$?

I can prove $\int_0^1 {\frac{1}{{{{(-\ln x)}^p}}}dx} $ converges when $p<1$, since $ - \log x \ge 1 - x$ when $x\in[0,1]$ and $\int_0^1 {\frac{1}{{{{(-\ln x)}^p}}}dx} < \int_0^1 {\frac{1}{{{{(1 ...
3
votes
1answer
47 views

How can I evaluate the following limit-integral combination?

Can you give me some hint on how to show that $$\lim_{y\to0^+}\frac{\int_0^\infty \exp(-y\cosh (x))\text dx}{\log y}=-1?$$ I tried to delimit from above and from below the function ...
1
vote
1answer
42 views

Testing for convergence. (Improper Integral)

How can I test this integral or convergence: $$ \int_1^\infty \frac{2x-1}{\sqrt{x^5 + 2x - 2}} dx $$ I'm trying to find integral of higher function and in result i get divergence, so I cant use this ...
-1
votes
1answer
39 views

Imroper integral. Show that this expresions are… [closed]

Show that: $$\int_0^\infty x^2e^{-x^2} \, dx = 1/2\int_0^\infty e^{-x^2} \, dx$$ How can i prove it? Can anyone help me.
0
votes
1answer
29 views

How do I show that the principal value of $\int_{- \infty}^{\infty}\sin(ax)\sin(bx)/x \,dx$ = 0

How do I show that the principal value of $\int_{-\infty}^{\infty}\sin(ax)\sin(bx)/x \,dx$ is equal to zero?
5
votes
4answers
177 views

Solve $\int \limits_{0}^{\infty} \frac{\cos(x)}{\cosh(x)} dx$ without complex integration.

Solve $$\int \limits_{0}^{\infty} \frac{\cos(x)}{\cosh(x)} dx$$ without complex integration. This integral can be very easily solved with contour integration, but how can you solve it without taking ...
1
vote
0answers
46 views

Integral involving exponential and powers of $x$

I am looking for the value of the following integral: $$\int_u^\infty \sqrt{(x^2-a)} \exp\left({-\left(bx^2+\frac{c}{x^2}\right)}\right)\text dx$$ I encountered this problem when trying to find the ...
3
votes
2answers
90 views

Is $\exp$ the only function satisfying $f(x)=\displaystyle \int_{-x}^{+\infty} f(-t) dt$?

Today in class we first dealt with improper integrals, and as an example we found $ \displaystyle \int_0 ^{+\infty} e^{-x}dx=1$. Soon, I noticed that in fact $$e^x=\int_{-x}^{+\infty}e^{-t}dt. $$ ...
2
votes
1answer
59 views

Help with this indefinite integral using residues?

Question: How to evaluate this integral using residues$$\int_{0}^{\infty} \frac{x \sin x}{1 + x^2} dx$$ I integrate over the entire real axis and dividing it by 1/2 since the integrand is even, ...
7
votes
3answers
530 views

How do I solve this improper integral: $\int_{-\infty}^\infty e^{-x^2-x}dx$?

I'm trying to solve this integral: $$\int_{-\infty}^\infty e^{-x^2-x}dx$$ WolframAlpha shows this to be approximately $2.27588$. I tried to solve this by integration by parts, but I just couldn't get ...
0
votes
3answers
52 views

Improper integral with discontinuity

Determine if $$\int_{0}^{\infty}\frac{e^{-1/x^2}}{x^2}dx$$ is convergent or not. Since the function is discontinous at $x=0$, I cannot apply comparison theorems for improper integrals. I have tried ...
1
vote
2answers
101 views

Closed expression of the following integral?

I believe that the following integral has a closed expression, but I haven't been able to check it $$I(k)=\int_{-\infty}^{\infty}dt\,\text{erf}\left(\frac{t}{b}-i \frac{1}{2}b(k+a)\right) ...
12
votes
5answers
365 views

Solve $\int_0^{\infty} {\sin(\tan(x)) \over x}dx$

I tried to solve it the Feynman way and defined: $$I(a):=\int_0^{\infty} {\sin(\tan(a \cdot x)) \over x} \ dx$$ And look what happens when one substitutes $u=ax$ $(a>0)$: $$I(a)=\int_0^{\infty} ...
0
votes
1answer
22 views

Improper Integral Converging or Diverging?

For the integral I = 1/(1+x)^e how do we known if it converges or diverges. Upper Limit of 0 and lower limit of -1. I know that it is improper - is it unbounded at x=-1? My understanding is that ...
2
votes
3answers
78 views

Evaluate the improper integral.

Evaluate the integral below. $ \int^{+\infty}_{-\infty} \frac{x^2}{{(x^2-8x+20)}^2} \, dx $ I feel that I know how to do this problem, but I'm getting caught up in all the calculations. I've been ...
1
vote
1answer
54 views

Integral of multplication of normal pdf and Rayleigh pdf distribution

I need to calculate following definite integral $$\frac{1}{2\pi }\int_0^\infty \frac{x^2 e^{-x^2/\sigma^2 } }{\sigma} \frac{e^{-\frac{\lambda}{{ax^2+b}}}}{\sqrt{ax^2+b}} ~~dx$$ It is actually ...
2
votes
1answer
42 views

Convergent or divergent?

$$\int_{-1}^0 \frac1{(1+x)^e} \,dx$$ I have tried using integral test and got an answer of converges with answer of $$\frac1{1-e}$$ is the correct? And also for $$\int_1^∞ e^{-2x} \, dx$$ that i ...
4
votes
2answers
57 views

compute improper integrals using integration by parts

Compute \begin{equation*} \int_0^\infty \frac{\sin^4(x)}{x^2}~dx\text{ and }\int_0^\infty \frac{\sin (ax) \cos (bx)}{x}~dx. \end{equation*} For the first integral I tried letting $u = \sin ^4 x$ ...
5
votes
2answers
72 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
35 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
24 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
67 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 ...
2
votes
1answer
59 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
35 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
31 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
70 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 ...
4
votes
3answers
374 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
30 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 ...
0
votes
2answers
29 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} ...
2
votes
3answers
66 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
14 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
63 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
31 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
84 views

Solve integral with exponent

How to solve integral: $$\int^\infty_0e^{-\frac{At^2}{t+1}}~dt , \quad A>0$$
2
votes
0answers
33 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
48 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 ...