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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89 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
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2answers
79 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.
6
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3answers
191 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
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3answers
99 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
75 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...): ...
8
votes
3answers
245 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 ...
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0answers
21 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
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3answers
38 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
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1answer
29 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
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4answers
50 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
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1answer
33 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
23 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
191 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 ...
6
votes
4answers
373 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 ...
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3answers
52 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: ...
3
votes
2answers
143 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
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2answers
34 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
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1answer
84 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
91 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
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7answers
132 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
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2answers
51 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?
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3answers
79 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
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1answer
64 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
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1answer
48 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
129 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
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2answers
45 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 ...
11
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3answers
126 views

parametric integral relating to hyperbolic function

Suppose that $a$ is real number such that $0<a<1$, how can we calculate $$ I(a)=\int_0^\infty \big(1-\frac{\tanh ax}{\tanh x}\big)dx .$$ As for some speical cases, I can work out $I(1/2)=1$. ...
0
votes
2answers
121 views

How to find $P(X>x)$ when the density is known but the integral does not seem to converge

I am trying to evaluate $$P(X>x) = \int_x^{\infty } t^{\kappa } \exp{\left(-\rho t^{\alpha\kappa + 1}\right)} \, dt$$ where $\kappa$, $\rho$ and $\alpha$ are all constants. I have tried some ...
6
votes
1answer
188 views

Closed form for $\int_1^\infty\frac{\operatorname dx}{\operatorname \Gamma(x)}$

Is a closed form for $$\int\limits_1^{+\infty}\frac{\operatorname dx}{\operatorname \Gamma(x)}$$known? I tried to find it, but all well-known integrals involving gamma-function (such as of ...
16
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1answer
339 views

Integral $\int_0^1\frac{\log(x)\log^2(1-x)\log^2(1+x)}{x}\mathrm dx$

I decided to follow a recent trend and ask a question about logarithmic integrals :) Is there a closed form for this integral? $$\int_0^1\frac{\log(x)\log^2(1-x)\log^2(1+x)}{x}\mathrm dx$$
0
votes
1answer
65 views

Problem 30 of GR8767 how is improper integral defined?

In the subject test of 1987 problem 30 asks: The improper integral $\displaystyle \int_a^b f(x) f'(x) \, dx$ is A.) necessarily zero B.) possibly zero but not necessarily C.) necessarily ...
13
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2answers
185 views

Evaluating $\int_{0}^{\pi/3}\ln^2 \left ( \sin x \right )\,dx$

Good evening! I want to compute the integral $\displaystyle \int_{0}^{\pi/3}\ln^2 \left ( \sin x \right )\,dx$. However I find it extremely difficult. What I've tried is rewritting it as: ...
1
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4answers
106 views

How would you integrate $\sqrt{1+\frac{1}{x^2}}$

I need to integrate $\sqrt{1+\frac{1}{x^2}}$ I've tried to let $u=1/x^2$ but end up with, $\int \frac{\sqrt{1+u^2}}{u^{3/2}}du$ . I attempted to then substitute $u=\tan\theta$ and lead me to ...
4
votes
3answers
99 views

Is $\int_{-1}^1 \frac{dx}{\sqrt{x^2 - 1}} $ divergent?

I would like to know if the following integral is divergent: $$\int_{-1}^1 \frac{dx}{\sqrt{1-x^2}} = \pi $$ Wolfram alpha returned a finite answer of $\pi$. It looks like it should have poles at ...
0
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1answer
38 views

An integral of Wolstenholme:$\int_0^{+\infty}\frac{\sum_1^n A_k\cos{a_k x}}{x}\mathrm {d} x$ where $\sum A_k=0$ and $a_k>0$

The book by Whittaker and Watson says it's equal to $-\sum_{k=1}^n A_k \log {a_k}$, and attibutes it to Wolstenholme. I believe this readily reduces to the simpler case of evaluating $\displaystyle ...
0
votes
3answers
39 views

Improper Integrals

Determine whether the following improper integral is convergent or divergent. $$\int_1^{\infty} \text{sech}\, x \ln x \,dx$$ I think that I need to use integration by parts but the sechx is really ...
1
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2answers
30 views

convolution problem given $h(x)=1/2$ for $0<x<2$ and $0$ otherwise

I have a convolution problem in the form $$g(x)= \int_{-\infty}^\infty h(y)h(x-y)\,dy$$ where they give me the function $h(x)=1/2$ for $0<x<2$ and $0$ otherwise. I have never done a ...
0
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1answer
47 views

When does this integral converge?

So I'd like to find out for which values of $a,b>0$ the following integral is well-defined and how that will change if the absolute value is removed? Thanks! $I = \int_0^\infty ...
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2answers
31 views

I need to find the formula for h(x) for all x

we're given a function $h(x)=\begin{cases}1/2&\text{for}&0\le x<2\\0 &\text{otherwise}\end{cases}$. Then we are told to define the function $\displaystyle g(x)= ...
1
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2answers
111 views

How to find the derivative of improper integral with variable upper limit?

I have the integral from $-\infty$ to $y^2$ of the function $(e^{-|x|})$ and I need to find the derivative of this. That is, $$\frac{d}{dy} \int_{-\infty}^{y^2} e^{-|x|}\,dx$$ Usually derivative ...
0
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3answers
48 views

Using the comparison test to evaluate $\int_1^\infty\frac{1}{1+x^2+16x^4}dx$?

So using the comparison test to evaluate $\int_1^\infty\frac{1}{1+x^2+16x^4}dx$, and we're given $\int_1^\infty\frac{1}{4x^2}dx$. So I have been trying to set up an inequality to use, but I can't seem ...
0
votes
2answers
108 views

Prove $f$ is uniformly continuous iff $ \lim_{x\to \infty}f(x)=0$

Let $f:[0,\infty)\to (0,\infty)$ be a continuous function and $\displaystyle\int^{\infty}_{0}f<\infty$. Show that $f$ is uniform continuous iff $\displaystyle\lim_{x\to \infty}f(x)=0$ So I ...
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4answers
105 views

Does the following integral converge: $\int_6^{\infty}\frac{dx}{\sqrt{1+x^2}}$

Does the following integral converge: $$\int_6^{\infty}\frac{dx}{\sqrt{1+x^2}}$$ I suppose we have to solve such problems by comparison test. All the integrals I tried so far do not fit the ...
1
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3answers
33 views

Finding the integral of $\int_{-\infty}^{\infty}e^{-|4x|}$.

So I am trying to find the integral of $\int_{-\infty}^{\infty}e^{-|4x|}$. I know the integral converges, and I know the answer as well, but I am confused on how to get the correct answer. My problem ...
1
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0answers
29 views

Integrating an expression over a vector $\mathbf{w}$

doing my homework for a Machine Learning course, I have to calculate the following expression: $\newcommand{\IDENTITY}{\mathbf{I}} \newcommand{\W}{\mathbf{w}} \newcommand{\WT}{\mathbf{w}^T} ...
23
votes
4answers
737 views

Closed-forms for several tough integrals

These integrals came up in the process of finding solution to Vladimir Reshetnikov's problem. I wonder if there are closed-forms for the following integrals: \begin{array}{1,1} &[\text{1}] ...
3
votes
0answers
17 views

Convergence of multiple integral in $\mathbb R^4$

Denote $(x,y,z,w)$ the euclidean coordinates in $\mathbb R^4$. I am trying to study the convergence of the integral $$\int \frac{1}{(x^2+y^2)^a}\frac{1}{(x^2+y^2+z^2+w^2)^b} dx\,dy\, dz\, dw$$ over a ...
5
votes
2answers
68 views

Integral with rational functions of powers and exponentials

Any ideas how to solve: \begin{equation} \int_0^\infty x^{n+\frac{1}{2}} (e^{a x }-1)^{-\frac{1}{2}} e^{i x t} dx \end{equation} where $a$ and $t$ are real, positive constants; $n$ is a positive ...
19
votes
3answers
317 views

Integral $\int_{0}^1\frac{\ln\frac{3+x}{3-x}}{\sqrt{x(1-x)}}dx$

I have a problem with the following integral: $$ \int_{0}^{1}\ln\left(\,3 + x \over 3 - x\,\right)\, {{\rm d}x \over \,\sqrt{\,x\left(\,1 - x\,\right)\,}\,} $$ The first idea was to use the ...