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
votes
3answers
57 views

Examine convergence of $\int_0^{\infty} \frac{1}{x^a \cdot |\sin(x)| ^b}dx$

Examine convergence of $\int_0^{\infty} \frac{1}{x^a \cdot |\sin(x)| ^b}dx$ for $a, b > 0$. There are 2 problems. $|\sin(x)|^b = 0$ for $x = k \pi$ and $x^a = 0$ for $x = 0$. We can write ...
3
votes
1answer
69 views

Improper Integral $\int _ {0}^{1} \frac{1}{\sqrt{(1-x) \sin{x}}} dx $

Does the following improper integral converges? $$\int _ {0}^{1} \frac{1}{\sqrt{(1-x) \sin{x}}} dx $$ I have tried some approaches but I'm not sure whether it was correct or not. First I split the ...
15
votes
1answer
661 views

Integrate this monster

Can you please help me? I've been trying for some time now to integrate this: $$\int_0^\infty g^{-(a+1)} \; \exp\left\{-\left(\frac{b}{g} + \frac{1}{2} \sum_{i=1}^{n} ...
0
votes
1answer
15 views

Show that $\int_{1}^{\infty}\frac{x-\left \lfloor x \right \rfloor -\frac{1}{2}}{x^{s+1}}(1-s\log x)\,\mathrm{d}x$ is convergent for $\Re s > -1$

I am struggling to understand the example in Special Functions p. 621, that states, that $$\int_{1}^{\infty}\frac{x-\left \lfloor x \right \rfloor -\frac{1}{2}}{x^{s+1}}(1-s\log x)\,\mathrm{d}x$$ is ...
0
votes
0answers
8 views

Proof: Majorant- and minorantcriterion for convergence of improper integrals

Let $I$ a interval and let $f,g:I\to \mathbb{R}$ continuous with $0\le g(x) \le f(x) \forall x \in I$. Prove this propostitions: (a) If $\int_{0}^{\infty} f(x)dx$ convergent $\Rightarrow$ ...
0
votes
1answer
46 views

A closed form of a integral with exp and cos

Can we find a closed form for the following integral: $$\int_0^{\infty} \frac{e^{-x} \cos x}{1+x} \, {\rm d}x$$ No matter how hard I tried I cannot tackle it. I am pretty much afraid that if a ...
1
vote
0answers
41 views

Evaluating a limit of an integral

I have a function $f(x,y,z) :\mathbb{R}^3 \rightarrow \mathbb{C}$, a smooth function. I know that $$ I = \int_{z \in \mathbb{R}}\int_{y \in \mathbb{R}}\int_{x \in \mathbb{R}} f(x,y,z) \ dx dydz $$ ...
6
votes
1answer
71 views

How was the difference of the Fransén–Robinson constant and Euler's number found?

I recently ran across the following integral: $$ \int_{0}^{\infty}\frac{1}{\Gamma(x)}dx $$ Which I learned is equal to the Fransén-Robinson constant. On the linked wikipedia page for the ...
1
vote
0answers
28 views

Integrating lower incomplete gamma function raised to the power $k$

I'm trying to solve the following integral: $$\int_0^\infty \gamma(t,x)^k x^t e^{-x} \mathrm{d} x$$ I'm fighting with it for quiet a while and didn't get any result. Though, I do have the ...
2
votes
4answers
113 views

Prove that $\int_0^1 \frac{\log{x}}{1-x^2}dx$ is convergent [on hold]

Could you please help me with proving that $$\int_0^1 \frac{\log{x}}{1-x^2}dx$$ is convergent?
1
vote
3answers
32 views

Showing the integral $\int_{\mathbb{R}} \int_{\mathbb{R}} \min\{ 1, (\max \{ |x|,|y| \})^{-3} \} dx dy$ converges

I am trying to bound the following integral: $\int_{\mathbb{R}} \int_{\mathbb{R}} \min\{ 1, (\max \{ |x|,|y| \})^{-3} \} dx dy$. I am very sure this integral converges, but whatever I try seem to ...
6
votes
3answers
115 views

Integrate $\int_0^\infty \frac{e^{-x/\sqrt3}-e^{-x/\sqrt2}}{x}\,\mathrm{d}x$

I can't solve the integral $$\int_0^\infty \frac{e^{-x/\sqrt3}-e^{-x/\sqrt2}}{x}\,\mathrm{d}x$$ I tried it by using Beta and Gamma function and integration by parts. Please help me to solve it.
5
votes
2answers
203 views

Calculate the Gauss integral without squaring it first

We know that the integral $$I = \int_{-\infty}^{\infty} \mathrm{d}x e^{-x^2}$$ can be calculated by first squaring it and then treat it as a $2-$dimensional integral in the plane and integrate it in ...
-1
votes
1answer
32 views

Integral convergence $\int_{2}^{\infty} \frac{\cos x}{\sqrt[3]{\ln x}}$

I've tried to partial this integral from 0 to 1, 1 to e, and e to infinity. $$\int_{2}^{\infty} \frac{\cos x}{\sqrt[3]{\ln x}} dx$$
0
votes
1answer
48 views

Are these integrals convergent?

Recently I've come across two integrals that seemed hard to check for me. Here they are: $$\int_0^\infty \frac{x \sin \ln x}{x^2 + \cos x} \, \mathrm{d}x$$ And another: $$\int_1^\infty \frac{\sin \ln ...
0
votes
1answer
44 views

Does the integral converge

How to prove that the following integral doesn't converge? $$\int_0^\infty \frac{1}{(\ln^4x + \ln^2x)\ln^2(1-x^{1/3})^2(x + \sqrt{x} + 1)}dx$$ I suppose it doesn't converge because of quick growth ...
8
votes
3answers
146 views

Need help with $\int_{-\infty}^\infty \frac{x^2 \, dx}{x^4+2a^2x^2+b^4}$

I'm having trouble trying to evaluate this definite integral. Mathematica didn't help much. $$\int_{-\infty}^\infty \frac{x^2 \, dx}{x^4+2a^2x^2+b^4}$$ where $a$, $b$ $\in \Bbb R^+$. Is it possible ...
18
votes
4answers
680 views

How to evaluate $I=\int_0^{\pi/2}\frac{x\log{\sin{(x)}}}{\sin(x)}\,dx$

Prima facie, this integral seems easy to calculate,but alas, this not's case $$I=\int_0^{\pi/2}\frac{x\log{\sin{(x)}}}{\sin(x)}\,dx$$ The numerical value is I=-1.122690024730644497584272... How to ...
0
votes
2answers
75 views

Is $\frac{1}{x^2}$ Lebesgue integrable while $\frac{1}{x}$ is not?

My textbook defined integrability as $f$ is said to be Lebesgue integrable if $\int{}f$ is finite. I heard that $\frac1x$ is not Lebesgue integrable, but $\frac{1}{x^2}$ is Lebesgue integrable. I do ...
37
votes
3answers
915 views

Find the value of $\int_0^{\infty}\frac{x^3}{(x^4+1)(e^x-1)}\mathrm dx$

I need to find a closed-form for the following integral. Please give me some ideas how to approach it: $$\int_0^{\infty}\frac{x^3}{(x^4+1)(e^x-1)}\mathrm dx$$
2
votes
2answers
37 views

Finding limit with improper integral

How should I approach this question? $$\lim_{x\to0}\frac{1}{x}\int_1^{1+x}\frac{\cos t}{t} \, dt$$ I tried to use L'hospital and that gave me $-\sin(0) = 0$ The correct answer is $\cos 1$. Did I ...
1
vote
0answers
51 views

unbounded solution, lim inf of trace,

Show that if $\lim \inf_{t\rightarrow \infty} \int_{t_0}^t \operatorname{tr}\left(A(s)\right)ds= \infty $ then the linear first-order system $x'(t)=A(t)x(t)$ where $A \in C\left(I, \mathbb{R}^{n\times ...
0
votes
1answer
18 views

“Nonlinear cosine” integral

Let $\alpha > 1$, $\xi \in\mathbb{R}$. and $\chi_A$ be the characteristic function of the set $A$. Are there some known ways of computing (or estimating in terms of $\xi$) of this kind of ...
1
vote
1answer
22 views

How to find the inverse Fourier transfmation of $\exp(-sk)/k$.

I've tried this with the help of hint given by one of my friend.He told me to first find the Inverse fourier transformation of $\exp(-sk)$ which is $$ \frac{\sqrt2}{\sqrt \pi}\frac{x}{x^2+ s^2}$$ ...
0
votes
1answer
46 views

Having trouble evaluting error function integrals

I am trying to evaluate $$I = \int_1^{\infty } \left(\frac{\operatorname{erf}\left(a -b\log (x)\right)}{2 x^2}-\frac{\operatorname{erf}\left(a + b\log (x)\right)}{2 x}\right) \, dx$$ Let $\log (x) = ...
1
vote
2answers
74 views

Deadly integral

How to solve this question $\int\limits_0^1\frac{x^{2}+x+1}{x^{4}+x^{3}+x^{2}+x+1}dx$ . Please help me in solving this short way my approach is in the answer Is it correct and can it be solved in ...
1
vote
0answers
26 views

Limit of improper integrals of uniformly convergent function

I've got a problem. Let $g(t)\ge0$ and it has improper integral on interval $[A, B)$. Furthermore, sequence of integrable functions $f_{n}(t)$ is uniformly convergent do $f(t)$ on every subinterval ...
3
votes
4answers
87 views

Limit of $ \frac1x \int_x ^{2x}e^{-t^2}dt$

What is the limit of the function $$\lim_{x\to 0} \ \frac1x \int_x ^{2x}e^{-t^2}dt$$ ? I tried this problem by using gamma function. I couldn't find the integral.
1
vote
2answers
44 views

Proving Euler's spiral is an isometric embedding with bounded image

$\newcommand{\al}{\alpha}$ I am trying to prove Euler's spiral is an isometric embedding of $\mathbb{R}$ into $\mathbb{R}^2$ with bounded image. Here is the definition of the spiral: $(*) \, ...
2
votes
1answer
81 views

Help evaluate $\int_0^\infty x\operatorname{erfc}(a + b\ln (x)) \,dx$.

I am trying to evaluate $$ I = \int_0^\infty x\operatorname{erfc}(a + b\ln (x)) \,dx $$ where $a \ge 0$ and $b> 0$. $$ I = \frac{2}{\sqrt{\pi}}\int_0^\infty \int_{a + b\ln (x)}^{\infty} ...
3
votes
2answers
207 views

Lebesgue Integral, existence, improper integrals, etc.

Problem: At the request of another user, I am taking an older question and specifically addressing one problem. I am self-learning about Lebesgue integration, and am just starting to try and apply ...
0
votes
1answer
83 views

Compute $\int_0^{\infty} \frac{1}{\sqrt{x^3 + x}}$

Could you tell me how to compute $$\int_0^{\infty} \frac{1}{\sqrt{x^3 + x}}dx$$ I have really no idea how to do this and I've tried for a couple of hours.
1
vote
1answer
32 views

Checking whther the integral $\int_1^∞ \frac{1}{x^{\frac{1}{x}+1}} dx$ convergent

I need to check convergence of $\int_1^∞ \frac{1}{x^{\frac{1}{x}+1}} dx$ . I think it divergence cause it bigger than $\int_1^∞ \frac{1}{x} dx$ but I can't prove it. I have an hint that ...
2
votes
2answers
44 views

Convergence and value of improper integral

I have to prove that integral $I = \int_{0}^{+\infty}\sin(t^2)dt$ is convergent. Could you tell me if it's ok? Let $t^2=u$ then $dt=\frac{du}{2\sqrt{u}}$ Now $$I = ...
0
votes
1answer
39 views

On a simple application of Paley-Wiener theorem and related doubts

Let $$F(x)=\frac{ \left\{ x \right\} }{e^{\sqrt{x}}},$$ be supported on $ \left( 0,\infty \right) $, where $ \left\{ x \right\} $ is the fractional part function. Then $F\in L^2(0,\infty)$ and the ...
5
votes
2answers
521 views

Calculate integral using beta and gamma functions

I have to calculate the following integral using beta and gamma functions: $$ \int\limits_0^1 \frac{x\,dx}{(2-x)\cdot \sqrt[3]{x^2(1-x)}} $$ I came up with this terrible solution. Firstly, let's ...
2
votes
4answers
90 views

Prove that $\int_0^{\infty} \frac{1}{\sqrt{x^3 + x}}dx$ is convergent

Could you tell me how to prove that $$\int_0^{\infty} \frac{1}{\sqrt{x^3 + x}}dx$$ is convergent?
10
votes
0answers
784 views

Integral of rational function over $\mathbb{H}^4$

Suppose I have a rational function of $8$ coordinates $a,b,c,d,e,f,g,h$ that I want to integrate over $\mathbb{H}^4$: ...
3
votes
1answer
66 views

Integral of $p(x)\operatorname{csch}(x)$

I'd like to calculate the following integral $$\int_{-\infty}^{+\infty}\frac{x^4 \left(\frac 1 {a^2+x^2} +\frac 1 {b^2+x^2}\right)}{\sinh^2(x\pi /c)} \, dx$$ where $a$, $b$ and $c$ are positive ...
1
vote
0answers
75 views

Evaluating $\int_0^\infty \sin x^2\, dx$ with real methods and without gamma functions?

I know that $$\int_0^\infty \sin x^2\, dx = \sqrt{\frac{\pi}{8}}$$ but all of the methods I've found seem to be too complicated for an early calculus student. Is there any method of calculating this ...
1
vote
2answers
39 views

Separation of integral by approximation

I'm working with the following integral $\displaystyle\int_0^y \frac{dx}{x \sqrt{1-ax-bx^2}}$ and would like to split it in something like $$\int_0^y\frac{dx}{x \sqrt{1-ax}}+\int_0^y\frac{dx}{x ...
1
vote
1answer
59 views

Infinite integral of $1/(1+x^2)$

Given the theorem that the infinite integral of $1/x^n$ is convergent if and only if $n>1$, I want to prove that the infinite integral of $1/(1+x^2)$ exists. This seems like a trivial question, I ...
20
votes
5answers
8k views

Prove: $\int_0^\infty \sin (x^2) \, dx$ converges.

$\sin x^2$ does not converge as $x \to \infty$, yet its integral from $0$ to $\infty$ does. I'm trying to understand why and would like some help in working towards a formal proof.
3
votes
5answers
93 views

Show that $\int_{-\infty}^{\infty}\frac{dx}{\sqrt{x^4+1}}$ converges.

Show that $$ \int_{-\infty}^{\infty}\frac{dx}{\sqrt{x^4+1}} $$ converges. I recognized that that since the integrand is even then $$ \int_{-\infty}^{\infty}\frac{dx}{\sqrt{x^4+1}} = ...
2
votes
3answers
75 views

Improper integral $\int _{0+0}^{1-0}\frac{dx}{\left(4-3x\right)\sqrt{x-x^2}}\:dx$

How do I solve this? $$\int _{0+0}^{1-0}\frac{dx}{\left(4-3x\right)\sqrt{x-x^2}}\:dx$$ I know it's a type 3 improper integral, and I'm having issues with these. I think that I need to write it as a ...
0
votes
1answer
46 views

How to solve this exp Integral

I am trying to solve the following integral, $$ I = \int_0^\infty \mathrm{e}^{z/2 - {\left(z - \ln a\right)^2}/4b} - \mathrm{e}^{z/2 - {\left(z + \ln a\right)^2}/4b}dz $$ where $a$ and $b$ are some ...
3
votes
3answers
83 views

Prove that $\int_0^{\infty} \frac{\log (1+x)}{x^2}dx$ is divergent.

Could you please tell me how to prove that $$\int_0^{\infty} \frac{\log (1+x)}{x^2}dx$$ is divergent? I calculated an indefinite integral but I don't know how to prove that it diverges.
1
vote
2answers
72 views

Convergence/divergence of a messy integral: $\int_1^\infty\frac{\arctan(9x)}{1+9x^3}\:dx$

Considering $$ \int_1^\infty\frac{\arctan(9x)}{1+9x^3}\:dx $$ I am trying to show convergence but looking to use Dirichlet's test and wanted to see if we can do it this way. Are we supposed to show ...
2
votes
3answers
121 views

Integral: $\int_{0}^{x}\lfloor\dfrac{1}{1-t}\rfloor dt$

I'm working on an integral problem(the rest of which is irrelevant) and this integral arises, which has stumped me. $$\int_{0}^{1}\int_{0}^{x}\left\lfloor\dfrac{1}{1-t}\right\rfloor dt dx$$ ...
-1
votes
2answers
72 views

$\int_0^{\infty} f(x)dx$ converges and $\lim_{x\rightarrow \infty} f(x)$ exists, then $\lim_{x\rightarrow \infty} f(x)=0$?

Is the following true or false: If $\int_0^{\infty} f(x)dx$ converges and $\lim_{x\rightarrow \infty} f(x)$ exists, then $\lim_{x\rightarrow \infty} f(x)=0$? This should be doable without series.