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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1answer
21 views

For what $\alpha$ does the integral absolutely and for what conditionally converge?

For what $\alpha$ does the integral absolutely and for what conditionally converge ? $$\int_{0}^{1}\frac{\ln^{\alpha} (1+x^4)}{x^4}\cos{1 \over x}dx$$ Not sure which criteria to use to prove the ...
1
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1answer
34 views

Let $a_n>0$ for $n \geq 1$ and let series: $\sum_{n=1}^{\infty}a_n$ diverge. Let $S_n=a_1+a_2+…+a_n > 1$ for $n \geq 1$

Prove that the series: $$\sum_{n=1}^{\infty}\frac{a_{n+1}}{S_n \ln S_n}$$ diverges and the series : $$\sum_{n=1}^{\infty}\frac{a_{n}}{S_n \ln^2 S_n}$$ converges. (Using the famous criteria I ...
1
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2answers
56 views

Find $f$ so that $\int_{1}^{\infty}f(x)dx$ exists, but $\displaystyle \lim_{x\to\infty}f(x)$ does not exist? [duplicate]

I need help finding an example of a function such that $\int_{1}^{\infty}f(x)dx$ converges, but $\displaystyle \lim_{x\to\infty}f(x)$ does not exist. I was trying to find examples of functions ...
1
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2answers
47 views

Given $|f(x)|=1$,how to construct an $f(x)$, such that $\int ^{+\infty }_{0}f\left( x\right) dx$ converges

Here's the problem: Given $|f(x)| = 1$, construct an $f(x)$, such that $$\int ^{+\infty }_{0}f\left( x\right) dx$$ converges. I think this problem may be done by dividing the 1s and -1s smartly, but ...
2
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0answers
78 views

Evaluate $\lim_{n\to\infty}\int_0^{\infty}\cos^n(x)dx$ [on hold]

How can I solve that $\lim_{n\to\infty}\int_0^{\infty}\cos^n(x)dx$?
1
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2answers
30 views

Convergence of a double integral

Is the integral $$\int_1^\infty\int_{e^{-x}}^1\frac{\sin y}{x^2y}dy dx$$ convergent or divergent?
3
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2answers
90 views

How to evaluate $\int_0^1 \ln(\frac{1+x}{1-x}) \frac{dx}{x} = \frac{\pi^2}{4}$?

Can anyone suggest the method of computing $\int_0^1 \ln(\frac{1+x}{1-x}) \frac{dx}{x} = \frac{\pi^2}{4}$ ? My trial is following first set $t =\frac{1-x}{1+x}$ which gives $x=\frac{1-t}{t+1}$ ...
1
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2answers
53 views

How to compute $\int_0^1 \frac{x-1}{\ln(x)} dx = \ln(2)$? and $\int_0^\infty \ln(t) e^{-t} dt $?

$\int_0^1 \frac{x-1}{\ln(x)} dx = \ln(2)$ First i try $\ln(x)=t$ so that $\frac{1}{x} dx =dt$ then integral becomes \begin{align} &\int_{-\infty}^{0}\frac{e^t-1}{t} (e^t dt) = - ...
7
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3answers
170 views

How to compute $\int_0^\infty \frac{x^4}{(x^4+ x^2 +1)^3} dx =\frac{\pi}{48\sqrt{3}}$?

$$\int_0^\infty \frac{x^4}{(x^4+ x^2 +1)^3} dx =\frac{\pi}{48\sqrt{3}}$$ I have difficulty to evaluating above integrals. First i try the subsititue $x^4 =t$ or $x^4 +x^2+1 =t$ but it makes ...
1
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1answer
35 views

Definite integral including natural log, cosine, and hyperbolic sine

Here is an integral question I have, I am solving some other problems like this but I am stumped on this one: $$\int_0^{\pi+1}\frac {\ln(\cos(x+1))}{\sinh(x^2)}dx$$ I used some methods such as ...
6
votes
0answers
31 views

Does such divergent integral assume the same values for any regularization?

Consider the integral: $$\int_0^\infty\sin(x)dx.\tag1$$ It's clearly divergent, but if we regularize it as $$\int_0^\infty\sin(x)e^{-x/a}dx=\frac{a^2}{a^2+1},\tag2$$ we can take the limit of ...
4
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0answers
68 views

How to find the value of this integral?

This integral to the value \begin{align} \int_0^1\frac{\ln^2(1+x)\ln^2 x}{1-x}\ dx=&\ ...
0
votes
2answers
33 views

Proving Integral Test?

Assume that $f(x) \geq 0$ and that $f$ decreases monotonically on $[1, \infty]$. Prove $\int_{1}^{\infty} f(x)dx$ converges iff $\sum_{n=1}^{\infty} f(n)$ converges. My proof: If $f$ is non-negative ...
0
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1answer
19 views

convergence of $\int_{x=0}^{\infty}x^{-\frac{M-1}{2}-N}(1-e^{-x})^{M-1}e^{-x}dx$

I am trying to study the convergence of $$\int_{x=0}^{\infty}x^{-\frac{M-1}{2}-N}(1-e^{-x})^{M-1}e^{-x}dx,$$ where $M$ and $N$ are positive integers. I've tried some $M$ and $N$, and it seems that ...
1
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0answers
26 views

The Laplace transform of $\exp(t^2)$

A naive attempt to calculate the Laplace transform of the function $f(t)=e^{t^2}$ results in integrals of the form $$\int_0^\infty e^{t^2-st}dt,$$ which obviously don't exist as the integrand grows ...
0
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1answer
42 views

Computing $\int_{0}^1 \frac{\left( (1-x )a +x b\right)^2}{(1-x)c +x d} dx$

I want to find the integral of \begin{align*} \int_{0}^1 \frac{\left( (1-x )a +x b\right)^2}{(1-x)c +x d}dx \end{align*} for any $a,b$ and $c>0$ and $d>0$. Using Wolfram-Alpha I found that ...
1
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1answer
54 views

Evaluate $\int_{x=0}^{\infty}\left(\frac{1}{\sqrt{x}}(1-e^{-x})\right)^{M-1}e^{-x}(1+sx)^{-N}dx,$

I am trying to evaluate $$\int_{x=0}^{\infty}\left(\frac{1}{\sqrt{x}}(1-e^{-x})\right)^{M-1}e^{-x}(1+sx)^{-N}dx,$$ where $s>0$, $M$ and $N$ are positive integers. But seem that the above integral ...
1
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1answer
47 views

Gamma and Beta function proof.

I'm trying to proof the equality $B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$ when $x,y>0,$ without using calculus in many variables. I've investigated about the topic but all references make ...
2
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2answers
34 views

Improper integral and lower Riemann sums

Given $f$ is positive and continuous on $(0,1]$ and its improper integral exists there. Is it true that the lower Riemann sums converges to the integral? I'm thinking about using definition but reach ...
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1answer
44 views

Improper Integral $e^{-x}/\sqrt x$ [closed]

How do I find $$\int^\infty _0 \frac{e^{-x}}{\sqrt x}$$
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1answer
21 views

Gamma and Beta Functions [closed]

\begin{equation*} \int \limits _0 ^\infty x^m \mathbb e ^{-x^n} \mathbb d x = \frac 1 n \Gamma (\frac {m+1} n), \space m>-1, \space n>0. \end{equation*} \begin{equation*} \int \limits _0 ^1 ...
0
votes
1answer
17 views

Infinite Integral of a Bessel Function

I need to calculate the following integral $$ \int_0^{\infty}xdxJ_n(kx) $$ Integrating it by parts and using the normalization of Bessel functions, I find it (somewhat heuristically) to equal the ...
4
votes
2answers
68 views

Show that $ \int_{-\infty}^{\infty} \frac{x^3}{(x^2+4)(x^2+1)}\, dx$ does not converge

I noticed that $\displaystyle \int_{-a}^{b} \frac{x^3}{(x^2+4)(x^2+1)}$ will converge to $0$ whenever $a=b$ and will converge to some value whenever $a,b$ are in the reals (excluding infinity). How ...
2
votes
2answers
26 views

explain the solution and/or suggest a different one

I have come across the following problem, in my calculus II course, about improper integrals: problem: Find the following limit, if it exists. $\displaystyle\lim_{x\to 1} \int\limits_{x}^{x^2} \! ...
1
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1answer
22 views

First order approximation of $F(x)=\int_0^x f(t) dt$ in the neighbourhood of $\infty$

Let $f(x)$ continuous on the real line. Then the first order approximation of $$F(x)=\int_0^x f(t) dt$$ in the neighbourhood of $0$ is: $$F(x)=\int_0^x f(t) dt\sim 0 + x f(0), \ \ \ (x\rightarrow 0)$$ ...
5
votes
2answers
101 views

How to compute $\int_{-1}^{1} e^{-1/(1-x^2)}dx$?

As in the title, I would like to compute the integral: \begin{equation} \int_{-1}^{1}e^{-1/(1-x^2)}dx \end{equation} My hunch tells me that I should try to transform it to the correspoding ...
0
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2answers
34 views

When using the Integral test, why is the value of the integral different from the sum of the series?

According to my textbook, the value of the improper integral is not always equal to the sum of the series. But why is that?
0
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1answer
45 views

Proving $\int_0^\infty e^{-ax}x^n\,dx = \frac{1}{a^{n+1}} \Gamma(n+1)$

Prove that $$ \int_{0}^{\infty} \ e^ {-ax} x^{n} dx = \frac{1}{a^{n+1}} \Gamma(n+1) \qquad (n>-1, \, a>0). $$ My try: Let $dv = e^{-ax}$ and $u = x^n$. Then $v = -\frac{1}{a}e^{-ax}$ ...
4
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3answers
63 views

The shortest way to prove that $\int_1^\infty \frac{{\arctan \left( x \right)}}{{\sqrt {{x^4} - 1} }}dx $ converges.

I'm trying to show that the integral $$\int_1^\infty \frac{{\arctan \left( x \right)}}{{\sqrt {{x^4} - 1} }}dx \quad \text{is convergent}.$$ We know that $$\frac{{\arctan \left( x \right)}}{{\sqrt ...
1
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1answer
71 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
254 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
42 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
68 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
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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
39 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
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2answers
54 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.
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votes
0answers
16 views

Help with improper integral (maybe hyperreal function).

I need proof: $f: [1, \infty) \to \mathbb{F}$ continuous, proof: $\displaystyle\int_{a}^{\to\infty}f(ax) dx$ is convergent, (eventually to $\infty$), $\forall a\geq 1$ and ...
0
votes
1answer
26 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
74 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 + ...
3
votes
2answers
72 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
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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
200 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
48 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
26 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
45 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
36 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
37 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
24 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
164 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 ...