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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1
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2answers
546 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
votes
3answers
54 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
131 views

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

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 ...
1
vote
4answers
116 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
vote
3answers
37 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
vote
0answers
33 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} ...
27
votes
4answers
929 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}] ...
4
votes
1answer
31 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
134 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
353 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 ...
27
votes
3answers
680 views

Integral of Combination Log and Inverse Trig Function

Does the following integral have a closed-form ?: \begin{equation} \int_{0}^{1}{\ln\left(\,x\,\right) \over 1 + x}\,\arccos\left(\,x\,\right) \,{\rm d}x \end{equation} This integral has been ...
4
votes
4answers
167 views

Integral $\int_{1}^{\infty} \frac{\log^3 x}{x(x-1)} dx$

How do I arrive at the closed form expression of the integral $$\displaystyle\int_{1}^{\infty} \dfrac{\log^3 x}{x(x-1)}dx$$ Most probably the closed form is $\dfrac{\pi^4}{15}$
0
votes
3answers
75 views

General closed form of $\int_{0}^{\theta_o} \dfrac{1}{\sqrt{\cos \theta-\cos \theta_o}} d\theta$

I once asked a question about how to integrate the reciprocal of the square root of cosine. Is there a general closed form for the integral $$\int_{0}^{\theta_o} \dfrac{1}{\sqrt{\cos \theta-\cos ...
7
votes
4answers
196 views

Calculus Question: Improper integral $\int_{0}^{\infty}\frac{\cos(2x+1)}{\sqrt[3]{x}}dx$

How to evaluate integral $$\int_{0}^{\infty}\frac{\cos(2x+1)}{\sqrt[3]{x}}dx?$$ I tried substitution $x=u^3$ and I got $3\displaystyle\int_{0}^{\infty}u \cos(2u^3+1)du$. After that I tried to use ...
3
votes
0answers
68 views

An integral with a decaying exponential with rational exponent

I was working on some mathematical derivations while I faced this integral: $$\Large \int_0^\infty x^{\alpha-1}e^{-\beta x} e^{-\lambda \left[\frac{x^2}{2x+\eta}\right]}\ \mathrm{d}x \quad .$$ Does ...
1
vote
3answers
53 views

Compute variance, using explicit PDF

I'm trying to get $\text{Var}(x)$ of $f(x) = 2(1+x)^{-3},\ x>0$. Please tell me if my working is correct and/or whether there is a better method I can use to get this more easily. $$ ...
0
votes
2answers
49 views

Help finding k. Issue with integration

Let the continuous random variable $X$ have a probability density function $f(x)$ such that $$f(x) = k(1+x)^{-3}, x>0$$ $=0$ elsewhere Find k This is what I tried: $\int_0^\infty k(1+x)^{-3}dx ...
6
votes
4answers
306 views

Wolfram alpha says that $\int_{-\infty}^\infty e^{-ix^2}dx = \sqrt{\frac{\pi}{i}}$

Wolfram alpha says that $$ \int_{-\infty}^\infty e^{-ix^2}dx = \sqrt{\frac{\pi}{i}}$$ holds. But it has two different values ($\sqrt{i}$). How should I understand this?
2
votes
3answers
56 views

Improper integral of rational function $k^2/(1+a^2k^2)^2$

I've got the integral $\int^\infty_{-\infty} dk \frac{k^2}{(1+a^2 k^2)^2}$ where $a$ is a real number. I can't seem to find a $u$-substitution or trigonometric substitution that will work. Any ...
4
votes
3answers
2k views

Deriving Mean and Variance of Laplace Distribution

It has been a long time since I have used calculus, and I am trying to understand how the mean and variance of the Laplace distribution with pdf $$f(x|\mu,\sigma) = \dfrac{1}{2 ...
0
votes
1answer
47 views

Solving Coulomb Integral in 1D

I am trying to solve the following Coulomb integral of two gaussians: $$ \int_{- \infty}^{ \infty}dx1\int_{- \infty}^{ \infty} \frac{e^{-b1 (x1-c1)^2}e^{-b2 (x2-c2)^2}}{\left | x1-x2 \right |}dx2, $$ ...
1
vote
1answer
51 views

Numerical integration of improper integral of the second kind

Peace be upon you, I had a simple and common question; but surprised after seeing no related results in search engine! I would like to know the numerical integration techniques for improper Riemann ...
4
votes
1answer
81 views

Calculus Question: Improper integral $\displaystyle\int_{-\infty}^{\infty} x^{2}e^{x-e^{2x}}dx$

I am curious about evaluation of the following integral $$\int_{-\infty}^{\infty} x^{2}e^{x-e^{2x}}dx$$ Is it possible to evaluate it? This not my homework but I will share my attempt. I tried ...
0
votes
0answers
31 views

Behavior of lower incomplete gamma function at complex infinity

The lower incomplete gamma function is given by $ \gamma\left(s, x\right) = \int\limits_0^x t^{s-1} e^{-t} {\rm d} t~,$ and has a well-defined analytic continuation for both $s$ and $x$ [1]. ...
2
votes
5answers
132 views

How to prove this integral?

I was confused about an integral showing on my teacher's slide, could anyone tell me how is the following integral derived? $$ \int^\infty_{-\infty} x^{2k} e^{-\frac{x^2}{2\sigma^2}} \; \mathrm{d}x ...
3
votes
0answers
96 views

How can I show that the integral equals zero?

Problem: Show that $$\int_{0}^{\pi / 2} \ln\left(\tan x - \sqrt{2 \tan x} + 1\right)\,\mathrm{d}x = 0 $$ I'd like to use, if possible, only single-variable Calculus methods, and it does not include ...
7
votes
2answers
387 views

Integral $\int\limits_0^1 \frac{\ln x\ln ( 1 - zx )}{1 - x} dx$

How can I evaluate following logarithmic integral: $$\int\limits_0^1 \frac{\ln x\ln ( 1 - zx )}{1 - x} dx$$
2
votes
2answers
85 views

Is there anything wrong with this proof?

$\lim_{n \to \infty} \int_{0}^{1} \dfrac{x^n}{1+x }dx=\lim_{n\to \infty} \xi^n \int_{0}^{1} \dfrac{dx}{1+x}=\lim_{n \to \infty} \xi ^n \ln{2}=(\ln{2}) \lim_{n \to \infty} \xi ^n =0 \qquad (0 \le \xi ...
2
votes
3answers
84 views

How to do this integral $\int_{-\infty}^{\infty}{\rm e}^{-x^{2}}\cos\left(\,kx\,\right)\,{\rm d}x$ [duplicate]

How to do this integral $$\int_{-\infty}^{\infty}{\rm e}^{-x^{2}}\cos\left(\,kx\,\right)\,{\rm d}x$$ for any $k > 0$ ?. I tried to use gamma function, but sometimes the series doesn't converge.
2
votes
2answers
86 views

$\int_0^{\infty}\frac{dx}{(x^2+\sqrt{x})^p}$ convergence

For which values of $p$ does integral $I$ converge? $$I=\int_0^{\infty}\frac{dx}{(x^2+\sqrt{x})^p}$$ My shot in the dark was $p\in(\frac{1}{2},2)$, since $\int_0^{1}\frac{dx}{x^{p/2}}<\infty \iff ...
10
votes
8answers
326 views

Evaluate $ \int_{0}^{1} \ln(x)\ln(1-x)\,dx $

Evaluate the integral, $$ \int_{0}^{1} \ln(x)\ln(1-x)\,dx$$ I solved this problem, by writing power series and then calculating the series and found the answer to be $ 2 -\zeta(2) $, but I don't ...
16
votes
5answers
723 views

Looking for closed-forms of $\int_0^{\pi/4}\ln^2(\sin x)\,dx$ and $\int_0^{\pi/4}\ln^2(\cos x)\,dx$

A few days ago, I posted the following problems Prove that \begin{equation} \int_0^{\pi/2}\ln^2(\cos x)\,dx=\frac{\pi}{2}\ln^2 2+\frac{\pi^3}{24}\\[20pt] -\int_0^{\pi/2}\ln^3(\cos ...
4
votes
2answers
165 views

$\int_{0}^{\infty} \frac{\cos(x)}{1+x^2} dx$ and $\int_{0}^{\infty} \frac {\ln(x)}{x^2+b^2} dx$

Prove that $$\int_{0}^{\infty} \frac{\cos(x)}{1+x^2} dx = \frac {\pi}{2e}$$ My approach would be $$\lim_{n \to \infty} \int_{0}^{n} \frac{\cos(x)}{1+x^2} dx$$ and evaluate the limits of the sine and ...
6
votes
4answers
291 views

Prove that $\int_0^{\pi/2}\ln^2(\cos x)\,dx=\frac{\pi}{2}\ln^2 2+\frac{\pi^3}{24}$

Prove that \begin{equation} \int_0^{\pi/2}\ln^2(\cos x)\,dx=\frac{\pi}{2}\ln^2 2+\frac{\pi^3}{24} \end{equation} I tried to use by parts method and ended with \begin{equation} \int \ln^2(\cos ...
32
votes
3answers
879 views

Prove $\displaystyle \int_{0}^{\pi/2} \ln \left(x^{2} + (\ln\cos x)^2 \right) \, dx=\pi\ln\ln2 $

How to prove $$ \int_{0}^{\pi/2}\ln\left(\,x^{2} + \ln^{2}\left(\,\cos\left(\,x\,\right)\,\right) \,\right)\,{\rm d}x\ =\ \pi\ln\left(\,\ln\left(\, 2\,\right)\,\right) $$ I don't know how to ...
5
votes
1answer
67 views

Convergence of Integral near 0

I am trying to determine the convergence of the integral \begin{equation} \int_0^1 \frac{f(x)}{x}\, dx \end{equation} given that $f(x)$ is bounded and continuous on $[0,1]$, and that $f(x)=0$. The ...
7
votes
1answer
150 views

A closed-form of $\frac{1}{2}\int_0^\infty\left[\frac{x^2\cos x}{\cosh 2x-\cos x}-\frac{2x^2}{e^{4x}-2e^{2x}\cos x+1}\right]\,dx$

I am looking for a closed-form of this integral \begin{equation} \frac{1}{2}\int_0^\infty\left[\frac{x^2\cos x}{\cosh 2x-\cos x}-\frac{2x^2}{e^{4x}-2e^{2x}\cos x+1}\right]\,dx \end{equation} I ...
1
vote
2answers
74 views

How to take the limit of the improper integral of a sequence of functions

Suppose $f_1, f_2, . . .$ are (Riemann) integrable functions. Then what is the $\epsilon$ definition of $$\lim_{n \rightarrow \infty} \lim_{M \rightarrow \infty} \int_{0}^{M} f_n(x) dx = L $$ for $L ...
0
votes
3answers
99 views

Value of convergence of $\displaystyle\int\frac{\sqrt{x}}{1+x^2}$ [closed]

How to prove that converge $$\int^{\infty}_1\frac{\sqrt{x}}{1+x^2}$$ and find this value.
8
votes
5answers
280 views
1
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0answers
23 views

Comparison of two integrals in $\Bbb R$

Is it possible to estimate $\int_{\mathbb{R}} |x|^2 u(x)\,\mathrm{d}x$ in terms of $\int_{\mathbb{R}} |x| u^2(x)\,\mathrm{d}x$ or estimate $\int_{\mathbb{R}} |x| u^2(x)\,\mathrm{d}x$ in terms of ...
0
votes
1answer
49 views

Convergence of $\int_0^{\infty}\left(\frac{x}{\ln x}\right)^c\frac{1}{1+x^{4c}}\ dx$

For which values of parameter $c\in\mathbb{R}$ is $$\int_0^{\infty}\left(\frac{x}{\ln x}\right)^c\frac{1}{1+x^{4c}}\ dx < \infty$$ Using the inequality $\ln x > \frac{x}{1+x}$ it can be shown ...
1
vote
2answers
49 views

For which values is this improper integral convergent?

I have a question here, which I would appreciate some help for: for which values of $\alpha$ is the improper integral $\int_0^1\frac{e^x - 1}{x^\alpha}dx$ convergent? I kind of get that I'm supposed ...
71
votes
6answers
5k views

How to find ${\large\int}_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx$

Please help me to find a closed form for this integral: $$I=\int_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx\tag1$$ I suspect it might exist because there are similar integrals having closed forms: ...
3
votes
2answers
96 views

cosine integral

Show that $$\int_0^x \frac{1-\cos(t)}{t}=\gamma+\ln(x)-\operatorname{Ci}(x)$$ where $$\operatorname{Ci}(x)=-\int_x^\infty \frac{\cos(t)}{t} \, dt$$ and gamma is an euler-mascheroni constant. I did as ...
4
votes
1answer
176 views

Intuitive reason for why the Gaussian integral converges to the square root of pi?

This is a very famous problem, which is commonly taught when students begin learning about multivariable integration in polar coordinates. However, it has always bothered me that we recieved such an ...
1
vote
4answers
79 views

How to prove that $\int_{0}^{\infty}{\frac{e^{-nx}}{\sqrt{x}}}\mathrm dx$ exists

I am trying to show that the integral $\int_{0}^{\infty}{\frac{e^{-nx}}{\sqrt{x}}}\mathrm dx$ exists ($n$ is a natural number). I tried to use the comparison theorem by bounding from above the ...
1
vote
1answer
43 views

How to prove that an integral converges

Let $(a_n)$, $(M_n)$ be sequences of positive real numbers such that ${a_n} \downarrow 0$, ${M_n} \uparrow \infty$ as $n\to\infty$. Let $\alpha>0$ and $\beta>1$. How to prove the following ...
8
votes
2answers
333 views

improper integral containing $\sqrt{\cos x-\dfrac{1}{\sqrt 2}}$ in the denominator

How do i find the value of this integral-- $$I=\displaystyle\int_{0}^{\pi/4} \frac{\sec^2 x \ dx}{\sqrt {\cos x-\dfrac{1}{\sqrt 2}}}$$ I came across this integral too in physics.
1
vote
4answers
198 views

Convergent or Divergent Integral

Convergent or Divergent? $$\int_0^1 \frac {dx}{(x+x^{5})^{1/2}} $$ I have problem with the fact that if we have integration from 0 to a say and a to infinity. How does this change the way we do ...