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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6
votes
2answers
74 views

Evaluating $\int_0^1 \frac{t^{a-1}}{1-t}-\frac{ct^{b-1}}{1-t^c}\ dt$

At first sight it looks like the integral below $$\int_0^1 \frac{t^{a-1}}{1-t}-\frac{ct^{b-1}}{1-t^c}\ dt$$ can be evaluated by using some geometric series. What else can we do? Is there a fast easy ...
0
votes
0answers
33 views

Is this integral in its most simplified form?

The following integration $$F(x)= \int_{x}^{+\infty} \frac{t}{1+t^\alpha} dt$$ cannot be solved in general, however can be expressed when $\alpha=4$ as $$F(x)= 0.5 \text{tan}^{-1} (x^{-2}) $$ it can ...
5
votes
3answers
117 views

Some integral representations of the Euler–Mascheroni constant

What kind of substitution should I use to obtain the following integrals? $$\begin{align} \int_0^1 \ln \ln \left(\frac{1}{x}\right)\,dx &=\int_0^\infty e^{-x} \ln x\,dx\tag1\\ &=\int_0^\infty ...
2
votes
1answer
23 views

Find the Values of $p$ and $q$ such that $\int_0^{\infty} x^pln(1+x)^q dx$ Converges or Diverges.

The question is to find the value values of $p$ and $q$ such that the improper integral converges or diverges. My friend indeed found some values of $p$ and $q$ such that it converges, but after many ...
2
votes
2answers
28 views

How to prove this multivariable integral identity?

By numerical experimentation I found that $$ \lim_{\beta \rightarrow \infty} \frac 1 \beta \int_0^{\beta}dx \int_0^{\beta}dy \, f\left( |x-y| \right) = 2\int_0^{\infty}dx \, f(x) $$ if $f:\mathbb{R} ...
19
votes
1answer
797 views

The Wicked Integral

My brother's friend gave me the following wicked integral with a beautiful result \begin{equation} {\Large\int_0^\infty} \frac{dx}{\sqrt{x} \bigg[x^2+\left(1+2\sqrt{2}\right)x+1\bigg] ...
2
votes
1answer
25 views

Integral test error approximation

Find an N so that: $\sum_{n=1}^\infty \frac{1}{n^4}$ is between $\sum_{n=1}^N \frac{1}{n^4}$ and $\sum_{n=1}^N \frac{1}{n^4} + 0.005$ I am getting N = 200. Is this correct? I don't know if I am ...
0
votes
1answer
14 views

inverse fourier transform of exponencial

Show that $F^{-1}(e^{-|x|}) =(\sqrt{2}/\sqrt{\pi})*1/(1+x^2)$ on $\mathbb R$. $F^{-1}$ is the inverse Fourier transform. Any help? how do you solve the integrals?
0
votes
1answer
15 views

Cauchy Principal Value different from the improper integral

I am having trouble formulating an example for which $\mathcal{P}\int^{\infty}_{-\infty}f(x)dx\neq\int^{\infty}_{-\infty}f(x)dx$ Would an example be $f(x)=1/x$ because of the asymptote at $x=0$?
3
votes
2answers
46 views

Prove that $\int_{0}^{+\infty} u^{s-1} \cos (a u) \:e^{-b u}\:du=\frac{\Gamma(s)\cos\left(s\arctan \left(\frac{a}{b}\right)\right)}{(a^2+b^2)^{s/2}}$

From the answer of this OP: Ramanujan log-trigonometric integrals, I found the following formula $$\begin{align} & \int_{0}^{+\infty} u^{s-1} \cos (a u) \:e^{-b u}\:\mathrm{d}u = \Gamma ...
3
votes
1answer
43 views

Why do we care about the 'rapidness' for convergence?

It is those puzzeling improper integrals that I can't get my head around.... Does the (improper) integral $\frac 1{x^2}$ from 1 to $\infty$ coverges because it is converging "fast" or because it has ...
1
vote
0answers
19 views

Determine whether $\lim_{R\to\infty}\int_0^R\frac{|\sin x|}{x}dx-\frac{2}{\pi}\ln R$ exists

Let $$J(R):=\int_0^R\frac{|\sin x|}{x}dx.$$ (i) Show that $$\lim_{R\to\infty}\frac{J(R)}{\ln R}$$ exists and determine its value (ii)Does $$\lim_{R\to\infty}J(R)-\frac{2}{\pi}\ln R$$ exist? If ...
6
votes
3answers
143 views

How to evaluate $\int_0^\infty \frac{1}{x^n+1} dx$ [duplicate]

Noticed that the integral $$\int_0^\infty \frac{1}{x^n+1} dx$$ is often approached with partial fraction decomposition, but this gets increasingly ugly as $n$ gets bigger. Is there a neat trick to do ...
0
votes
1answer
15 views

Tempered distribution and primitive integral

$f$ is a Schwartz function on $\mathbb{R}$. Define $g(x)= \int_{-\infty}^{x} f(x)dx$. Show that $g(x)$ is a tempered distribution. Any ideas? I have no idea how to do the problem
1
vote
0answers
16 views

Is it always true that an integral diverges on (a,b) if it diverges on (c,b) and a<c<b?

I know that the integral of 1/x diverges on (0,b). I know the same integral still diverges on (-b, b), despite the fact it seems it should be zero. My question then is, "Is it ALWAYS the case that ...
3
votes
1answer
63 views

Evaluating the integral $ \int_0^{\infty} \cos(x^2)\, \mathrm{d} x$?

Is it necessary to make use of the Gaussian integral and the complex exponential form of the cosine in evaluating the following integral? $$\int_0^{\infty} \cos(x^2)\, \mathrm{d} x$$ Just curious - ...
1
vote
1answer
55 views

Evaluate the integral if possible

Evaluate the following integral, if possible: $$\int_1^4 \frac{w}{w-3}dw$$ $$u= w-3,\; du = dw$$ $$\int_1^4 \frac{u+3}{u}du \Rightarrow \int_1^4 \frac{u}{u}du + \int_1^4 \frac{3}{u}du$$ ...
2
votes
0answers
50 views

difficult integral [duplicate]

Evaluate the following integral : $\int _0^{\infty }\:\left(\frac{^{\:}\sin \left(x\right)}{x}\right)^3dx$
1
vote
2answers
53 views

Evaluate the integral $\int_{1}^{4} w/(w-3) dw$ if possible

Evaluate the following integral, if possible: $$\int_{1}^{4} \frac{w}{w-3} dw.$$ I am supposed to be using improper integrals so I know I should find $$ \lim_{t \rightarrow3^-} \int_{1}^{t} ...
10
votes
1answer
151 views

Two integral involving logarithm and polylogarithm function

Evaluate the following integrals $$\int_0^1\frac{\ln(1-x)}{x}\text{Li}_3\left(\frac{1 + x}{2} \right)\,dx\\ .\\ \int_0^1\frac{\ln^2(1-x)}{x}\text{Li}_2\left(\frac{1 + x}{2} \right)\,dx$$
1
vote
1answer
36 views

Check the convergence (& absolutely) of parametric integral [closed]

$$\int\limits_{-1}^{1} \left(\frac{1+x}{1-x}\right)^{\alpha} ln(2+x)dx$$ Don't know where to start..
1
vote
1answer
38 views

Calculating improper integral

Does anyone know how to solve the following integral: $$I =\int_{0}^\infty \cos(t \mathrm{log}( x))\,\mathrm{e}^{-ax}\, \mathrm{d}x,$$ where $t$ and $a$ are real. Please show some intermediate ...
5
votes
2answers
93 views

About Integration

How to calculate the following integral $$ \int \frac{\tanh(\sqrt{1+z^2})}{\sqrt{1+z^2}}dz $$ Is there any ways to calculate those integral in analytic? (Is $[0,\infty]$, case the integral is ...
1
vote
0answers
53 views

Proof of Definite Integral of Even Function for Improper Integrals

I am trying to prove $\displaystyle \int_{\mathop \to -a}^{\mathop \to a} f \left({x}\right) \ \mathrm d x = 2 \int_0^{\mathop \to a} f \left({x}\right) \ \mathrm d x$ for $f$ which is an even ...
2
votes
1answer
61 views

Calculating a complex definite improper integral: $I= \int_{0}^\infty x^{it}\,\mathrm{e}^{-ax}\, dx$

Does anyone know how to find the value of this integral: $$I= \int_{0}^\infty x^{it}\,\mathrm{e}^{-ax}\, dx,$$ where $i=\sqrt{-1}$ and $t$, $a$ are real. Please give me a hint. Thank you.
10
votes
3answers
141 views

Evaluate $\int_{0}^{\infty} \mathrm{e}^{-x^2-x^{-2}}\, dx$

I have to find $$I=\int_{0}^{\infty} \mathrm{e}^{-x^2-x^{-2}}\, dx $$ I think we could use $$\int_{0}^{\infty} \mathrm{e}^{-x^2}\, dx = \frac{\sqrt \pi}{2} $$ But I don't know how. Thanks.
7
votes
3answers
90 views

Calculate trigonometric integral $ \int_{-\infty}^{\infty}{\sin(x^2)}\,dx$

Recently, I came across the following integral: $$ \int_{-\infty}^{\infty}{\sin(x^2)}\,dx=\int_{-\infty}^{\infty}{\cos(x^2)}\,dx=\sqrt{\frac{\pi}{2}} $$ What are the different ways to calculate such ...
4
votes
2answers
69 views

How to compute $\int_0^{\infty} x^{t-1} e^{-x}\ln(x)\,dx$?

I have hit the following integral (in the process of trying to derive a finite-sample correction for the Maximum Likelihood fitting of the Generalized Extreme Value distribution...): ...
6
votes
3answers
152 views

Evaluate $\int^1_0 \log^2(1-x) \log^2(x) \, dx$

I have no idea where to even start. WolframAlpha cant compute it either. $$\int^1_0 \log^2(1-x) \log^2(x) \, dx$$ I think it can be done with series, but I am not sure, can someone help a little so ...
1
vote
0answers
12 views

How naturally can functions defined by parametric integrals be interpolated from $\mathbb N$ to $\mathbb R^+$?

This question has not got any answer on Mathoverflow. I admit that it is unusual to cross-post in this direction (from MO to math.SE), but knowing that some of those really unbelievable integrals tend ...
1
vote
3answers
35 views

Integral convergence and limit question

I have a question that has come up in a group homework project that neither I nor my partner have any idea how to solve. I'm hoping someone can give me a hint or some guidance as to how to go about ...
0
votes
0answers
18 views

convergence of sequence of functions with finite second moment

Given $0<a<1$. Let $\phi:\mathbb R\mapsto\mathbb R$ is defined by $\phi(x)=\frac1{\sqrt{2\pi}}e^{-\frac{x^2}2}$ for all $x\in\mathbb R$. Suppose we are given a sequence of functions $\{f_n\}$ ...
0
votes
1answer
27 views

Convergence uniformly implies in integral

Let $\phi:\mathbb R\mapsto\mathbb R$ is defined by $\phi(x)=\frac1{\sqrt{2\pi}}e^{-\frac{x^2}2}$ for all $x\in\mathbb R$. Suppose we are given a sequence of functions $\{f_n\}$ such that ...
3
votes
4answers
48 views

What do limits of functions of the form $te^t$ have to do with l'Hopital's rule?

I have an improper function that I have to integrate from some number to infinity. Once integration is done, the function is of the form $te^t$. What I'm wondering is what does this have to do with ...
1
vote
1answer
27 views

Solve $\int\limits_{-\infty}^{\infty}e^{-cx^2}\sin(sx)dx $

How to prove that $$\int\limits_{-\infty}^{\infty}e^{-cx^2}\sin(sx)dx = 0,$$ where $c>0$?
1
vote
1answer
19 views

Integral solution using modified Bessel function of order 1

How to solve this integral making use of Modified Bessel function of order one? $\frac{q\gamma b}{2\pi}\int_{-\infty}^{\infty}(\gamma ^{2}\nu ^{2}t^{2}+b^{2})^{-3/2}\exp (i\omega t)dt$ Context: I ...
5
votes
3answers
170 views

Integral of greatest integer function divided by an exponential

If $\lfloor x \rfloor$ denotes the greatest integer not exceeding $x$, then find $\displaystyle\int_{0}^{\infty} \displaystyle \frac{\lfloor x \rfloor}{e^{x}} dx$. The correct answer is supposed to be ...
5
votes
4answers
239 views

How to compute $\int_{-\infty}^\infty\exp\left(-\frac{(x^2-13x-1)^2}{611x^2}\right)\ dx$

$$\int_{-\infty}^\infty\exp\left(-\frac{(x^2-13x-1)^2}{611x^2}\right)\ dx$$ WolframAlpha gives a numerical answer of $43.8122$, which appears to be $\sqrt{611\pi}$. And playing with that, it seems ...
1
vote
3answers
42 views

Find the Values of $p$ and $q$ Such that the Improper Integral Converges

I have a partial solution only so far. The integral is $$\int_0^{1/2} x^p(-\ln x)^q dx$$ With the substitution $-\ln x = y$, then $u = (p+1)y$, we get something like the gamma function: ...
2
votes
2answers
108 views

Evaluate $\int_{0}^{\frac {\pi}{3}}x\log(2\sin\frac {x}{2})\,dx$

Prove $$\int_0^{\pi/3}x\log \left(2 \sin\frac {x}{2}\right)\,dx = \frac {2\zeta(3)}{3}-\frac {\pi^2}{9}\log (2\pi)+\frac {2\pi ^2}{3}\log \left|\frac {\Gamma_2 \left(\frac {5}{6}\right)}{\Gamma_2 ...
1
vote
2answers
33 views

What will happen after Laplace Transform?

Consider the Laplace transform $\int_{0}^{\infty} e^{-px}f(x)\,dx$ Assume $f(x)=1$ , then the Laplace transform is $\frac {1}{p}$. Assume $f(x)=x$ , then the Laplace transform is $\frac {1}{p^2}$. ...
4
votes
1answer
65 views

Electrostatic Potential Energy integral in spherical coordinates

I'm having trouble with evaluating an integral that arises from attempting to find the total energy of an electrostatic system consisting of two point charges, which involves an integral over all ...
6
votes
0answers
75 views

Solving a problem of Ramanujan's interest

I am Brian Diaz, and I am new to the math.stackexchange community. I have been struggling with attempting to find a closed form of the following series: $$ \varphi(\theta) = 1 + 2\sum_{n=1}^{\infty} ...
6
votes
7answers
124 views

How to integrate $\int_{-\infty}^\infty e^{- \frac{1}{2} ax^2 } x^{2n}dx$

How can I approach this integral? ($0<a \in \mathbb{R}$ and $n \in \mathbb{N}$) $$\large\int_{-\infty}^\infty e^{- \frac{1}{2} ax^2 } x^{2n}\, dx$$ Integration by parts doesn't seem to make ...
0
votes
2answers
50 views

The integral $\int_0^\infty f(x)g(x) e^{-x}\,dx$ is convergent for all real polynomials $f,g$ [duplicate]

For all real polynomials $f$, $g$ Why is the integral $\int_0^\infty f(x)g(x) e^{-x}\,dx$ convergent?
-1
votes
3answers
60 views

Calculus find extreme values of integral

I have this problem: Ineach of Exercises 48-51, a definite integral is given. Do not attempt to calculate its value $V$. Instead, find the extreme values of the integrand on the interval of ...
0
votes
1answer
36 views

Integration by parts on all of $\mathbb{R}^n$ with $n>1$

So this came up as I was thinking about the uniqueness of solutions to the wave equation. I have seen proofs for uniqueness on all of $\mathbb{R}$ or on bounded subsets of $\mathbb{R}^n$, but never ...
0
votes
1answer
36 views

Proving an integration with a modified Bessel function and an exponential

I am trying to prove the following identity: where $\mu, h, H$, and $\tilde{\gamma}$ are real constants. The only hint that I have is use the relation between the modified bessel function of the ...
2
votes
3answers
128 views

Help me with this definite integral

I don't know how to solve this definite integral, maybe the solution is evident but i don't see it : $\int_0^\frac{\pi}{2} \frac{\cos^3(x)}{(\cos(2x) + \sin(x))}\,dx$
0
votes
2answers
41 views

Integrating this improper integral to test for convergence?

I'm trying to integrate this: $$\int^\infty_0 \frac{8}{\sqrt{e^{x}-x}} \,dx$$ And use the Direct Comparison Test to find out whether it diverges or converges. I looked at a similar problem: and I ...