Tagged Questions
4
votes
1answer
60 views
Test of convergence of $\int_{-\infty}^{\infty} \dfrac{x^6+6}{x^8+8}dx$
I am having some trouble with this problem and don't know if I am doing it right:
$$\int_{-\infty}^{\infty} \dfrac{x^6+6}{x^8+8}dx$$
so the steps I have taken so far are, I split it into
...
2
votes
2answers
45 views
How to calculate $\lim_{x\to 1}\int_{0}^{1}\frac{dy}{\sqrt{1-y^{2}}}\frac{y^{3/2}}{\sqrt{x - y}}$ when $x>1$?
Numerically, it looks that the limit is
$$\lim_{x\to 1}\int_{0}^{1}\frac{dy}{\sqrt{1-y^{2}}}\frac{y^{3/2}}{\sqrt{x - y}} = \frac{1}{\sqrt{2}}\log(1 - x) + cte $$,
but I have not been able to ...
5
votes
2answers
124 views
Trig Fresnel Integral
$$\int_{0}^{\infty }\sin(x^{2})dx$$
I'm confused with this integral because the square is on the x, not the whole function. How can I integrate it? Thank you.
I have not done complex analysis (only ...
6
votes
1answer
77 views
integration as limit of a sum
If $f$ is continuous on $[0, 1]$ then
$$\lim_ {n\to\infty}\sum_{j=0}^{\lfloor n/2\rfloor} \frac1{n}f\left(\frac {j}{n}\right) = ? $$
will the answer be that the limit exists and is equal to $ ...
1
vote
1answer
95 views
Proof that Riemann-integrability is preserved by changing the function at a converging sequence of points
Let $(x_n)^{\infty}_{n=1}\in[a,b]$ such that $\lim\limits_{n\to\infty}x_n=l$. Let $g:[a,b]\to \mathbb{R}$ be bounded function. Assume that there exists a Riemann integrable function $f:[a,b]\to ...
14
votes
1answer
155 views
How to prove $\frac{\pi^2}{6}\le \int_0^{\infty} \sin(x^{\log x}) \ \mathrm dx $?
I want to prove the inequality
$$\frac{\pi^2}{6}\le \int_0^{\infty} \sin(x^{\log x}) \ \mathrm dx $$
There are some obstacles I face: the indefinite integral cannot be expressed in terms of ...
0
votes
0answers
106 views
Riemann sums vs Darboux sums
Let we speak of the tagged partitions of an interval and a bounded function defined on it.
I think that the tags give rise to particular Riemann sums which may be quite different in value that the ...
0
votes
0answers
23 views
An integral inequality related to Taylor expansion
Problem. Let $f:[a,b]\to\mathbb{R}$ be a function such that $ f\in C^3([a,b])$ and $f(a)=f(b)$. Prove that $$ ...
2
votes
2answers
73 views
Asymptotic expansion for $\frac{1}{2\zeta(3)}\int_x^\infty \frac{u^2}{e^u - 1} du$?
Is there an asymptotic expansion for the function:
\begin{equation}
g(x)=\frac{1}{2\zeta(3)}\int_x^\infty \frac{u^2}{e^u - 1} du,
\end{equation}
over the domain $x\in [0,\infty)$ in terms of ...
2
votes
1answer
47 views
Further explanation needed for “If $f_n(x) \longrightarrow 0$ almost everywhere (a.e.) and…”
I came across the following problem and do not know how to proceed:
Let $\{f_n\}$ be a sequence of integrable functions defined on an interval $[a,b]$. Then I have to prove that
"If ...
10
votes
2answers
190 views
How to calculate $\int_{0}^{1}(\arcsin{x})(\sin{\frac{\pi}{2}x})dx$?
How to find the follwing integral's value ?
$$\int_{0}^{1}(\arcsin{x})(\sin{\frac{\pi}{2}x})dx$$
Actually, I don't know it can be represented as closed form.
0
votes
0answers
30 views
Fundamental theorem of calculus 1 where integrand is a 2nd order partial derivative
I have a function $b(x,y)$ such that $b(x,0)=0$.
Now, suppose I wish to evaluate the following integral: (Note that $b$ is continuous almost everywhere but it is assumed that it is integrable. Also, ...
11
votes
3answers
352 views
The equivalence between Cauchy integral and Riemann integral for bounded functions
Definitions
Suppose $P\colon a=x_0<x_1<\dotsb<x_n=b$ is a partition of $[a,b]$. Let $\Delta x_k=x_k-x_{k-1}$ and $\lVert P\rVert$ denotes $\max_{0<k\le n}\Delta x_k$.
The Cauchy integral ...
10
votes
6answers
418 views
Alternative solutions to $\lim_{n\to\infty} \frac{1}{\sqrt{n}}\int_{ 1/{\sqrt{n}}}^{1}\frac{\ln(1+x)}{x^3}\mathrm{d}x$
Here is a limit that can be computed directly by performing the integration and then taking
the limit, but the way is rather ugly. What else can we do? Might we avoid the integration?
...
0
votes
1answer
31 views
A small clarification
Could you please explain the following result?
$$n\left(\frac{1}{n}\sum_{i=1}^nf \left(\frac{i}{n}\right)-\sum_{i=1}^n \int^\frac{i}{n}_\frac{i-1}{n}f(x)dx ...
0
votes
2answers
132 views
Riemann Integral
Q. Compute the following limit:
$$\lim_{n\rightarrow\infty}\left(\frac{2^{1/n}}{n+1}+\frac{2^{2/n}}{n+1/2}+\cdots+\frac{2^{n/n}}{n+1/n}\right)$$
using integral.
One obvious method is to squeeze it ...
2
votes
1answer
126 views
A multiple integral question II
We know from the previous post that
$$\lim_{n\to\infty}\underbrace{\int_0^1 \int_0^1 \cdots \int_0^1}_{n \text{ times}}\frac{1}{(x_1\cdot x_2\cdots x_n)^2+1} ...
6
votes
1answer
90 views
A multiple integral question
Proving that
$$\lim_{n\to\infty}\underbrace{\int_0^1 \int_0^1 \cdots \int_0^1}_{n \text{ times}}\frac{1}{(x_1\cdot x_2\cdots x_n)^2+1} \mathrm{d}x_1\cdot\mathrm{d}x_2\cdots\mathrm{d}x_n=1$$
1
vote
1answer
85 views
limit of the error in approximating definite integral with midpoint rule
I want to calculate $\lim_{n \rightarrow \infty} n^2 |\int_{[0,1]}f(x)-I_n(x)|$ where $I_n$ is the integral approximation by midpoint rule:
$I_n=\frac{1}{n}\sum_{k=1}^nf(c_k)$ and $c_k$ is the point ...
4
votes
2answers
146 views
Evaluation of $\lim_{n\to\infty} \int_0^1 \frac{e^{\displaystyle x^{n}}}{1+x^2}\,\mathrm{d}x$
Evaluation of
$$\lim_{n\to\infty} \int_0^1 \frac{e^{\displaystyle x^{n}}}{1+x^2}\,\mathrm{d}x$$
Sis.
13
votes
2answers
275 views
Evaluation of $\int_0^{\pi/3} \ln^2\left(\frac{\sin x }{\sin (x+\pi/3)}\right)\,\mathrm{d}x$
I plan to evaluate
$$\int_0^{\pi/3} \ln^2\left(\frac{\sin x }{\sin (x+\pi/3)}\right)\, \mathrm{d}x$$
and I need a starting point for both real and complex methods. Thanks !
Sis.
2
votes
2answers
119 views
How to evaluate a definite integral that involves $(dx)^2$?
For example: $$\int_0^1(15-x)^2(\text{d}x)^2$$
2
votes
0answers
89 views
Integral representation of Euler's constant
Prove that : $$ \gamma=-\int_0^{1}\ln \ln \left ( \frac{1}{x} \right) \ \mathrm{d}x.$$
where $\gamma$ is Euler's constant ($\gamma \approx 0.57721$).
This integral was mentioned in Wikipedia as in ...
9
votes
1answer
70 views
Prove that $\int_a^cf(x)\mathrm{d}x+(c-a)g(c)=\int_c^bg(x)\mathrm{d}x+(b-c)f(c)$
Let $f$ , $g$ be real continuous functions in $[a,b]$. Prove that there is $c\in(a,b)$ such that
$$\int_a^cf(x)\mathrm{d}x+(c-a)g(c)=\int_c^bg(x)\mathrm{d}x+(b-c)f(c)$$
What would you suggest me to ...
8
votes
4answers
194 views
Calculate : $\int_1^{\infty} \frac{1}{x} -\sin^{-1} \frac{1}{x}\ \mathrm{d}x $
Find : $\displaystyle \int_1^{\infty} \frac{1}{x} -\sin^{-1} \frac{1}{x}\ \mathrm{d}x $.
I've done some work but I've got stuck, you may try to help me continue or give me another way , in both cases ...
2
votes
1answer
124 views
Characteristic function of the Smith-Volterra-Cantor set
Let the characteristic function of the SVC set be denoted by $ \beta $. Does the Riemann integral $ \displaystyle \int_{0}^{1} \beta ~ d{x} $ exist? I think it does since $ \beta $ is bounded, but I ...
1
vote
3answers
57 views
A rational function integral
How to evaluate :
$$\int_{0}^{1}{\frac{{{x}^{a-1}}}{1+{{x}^{b}}}}\text{d}x,\ \ \ a,\ b\in {{\mathbb{N}}^{+}}$$
5
votes
3answers
90 views
Compute the following limit $ I=\lim_n\int_{[0,1]^n}f(x)\,dx_1\ldots dx_n$. $f$ being the max function over $n$ coordinates.
Compute the following limit
$$
I=\lim_n\int_{[0,1]^n}f(x)\,dx_1\ldots dx_n,
$$
where
$$
f:\mathbb{R}^n \to \mathbb{R},\ x=(x_1,\ldots,x_n) \mapsto \max\{x_1,\ldots,x_n\}.
$$
30
votes
2answers
604 views
Prove that $\int_0^{\infty} \frac{\sin(2013 x)}{x(\cos x+\cosh x)}dx=\frac{\pi}{4}$
Prove that
$$\int_0^{\infty} \frac{\sin(2013 x)}{x(\cos x+\cosh x)}dx=\frac{\pi}{4}$$
1
vote
2answers
72 views
A integral with hyperbolic function
Evaluate :
$$\int_{0}^{2{\rm arccosh\,} \pi }{\frac{\text{d}x}{1+\frac{{{\sinh }^{2}}x}{{{\pi }^{4}}}}}$$
1
vote
0answers
58 views
Proof of $f=0$ a.e. in $[a,b]$ then $f$ is gauge integrable and $\int_a^bf=0$
Let $f:[a,b]\to \mathbb{R}$ so that $f(x)=0$ almost everywhere in $[a,b]$. Prove that $f$ is gauge integrable and $\int_a^bf=0$.
How can this be proven using the following definition of measure: ...
4
votes
0answers
86 views
Mistake in Bartle's proof of Hake's Theorem?
Here is Bartle's proof of Hake's Theorem found in "A Modern Theory of Integration". I think there is a mistake in the highlighted line:
The Theorem: $f:[a,b]\to \mathbb{R}$ is gauge integrable if and ...
2
votes
0answers
79 views
Proof that if $f,g,h:[a,b]\to \mathbb{R}$ with $h\le f,g$ and $f,g,h$ are gauge integrable then so is $\min(f,g)$
I am asking for a self contained proof of this assertion:
If $f,g,h:[a,b]\to \mathbb{R}$ with $h\le f,g$ and $f,g,h$ are gauge integrable then so is $\min(f,g)$.
The integral in question is the ...
8
votes
1answer
149 views
Compute $\lim_{s\to 0} \left(\int_0^1 (\Gamma (x))^s\space\mathrm{dx}\right)^{1/s}$
Compute
$$\lim_{s\to 0} \left(\int_0^1 (\Gamma (x))^s\space\mathrm{dx}\right)^{1/s}$$
This is a problem I thought of these days and I think I know a way although
not
completely justified. This is ...
12
votes
2answers
234 views
Finding $\int^1_0 \frac{\log(1+x)}{x}dx$ without series expansion
I was trying to evaluate $$\int^1_0 \frac{\log(1+x)}{x}dx.$$
I expanded $\log(1+x) $ as
$x -\frac{x^2}{2}... $ and got the answer. I would like to know if there is any way to do it without series ...
2
votes
4answers
100 views
prove or disprove that $\int_a^b |f(x)| \mathrm{d}x\geq |\int_a^b f(x)\mathrm{d}x |$
+Let $f$ be a continuous and integrable function over $[a;b]$, Prove or disprove that :
$\displaystyle\int_a^b |f(x)|\ \mathrm{d}x\geq \left | \int_a^b f(x)\ \mathrm{d}x\right|
$
1
vote
0answers
53 views
Proof that if $f$ is integrable then so is $\left|f\right|$ with the classical definition of the Riemann Integral
Let $f:[a,b]\to \mathbb{R}$ be integrable. Prove $\left|f\right|$ is integrable with the definition of the integral using the Riemann sums (not the Darboux one).
I think the way to go is to use the ...
1
vote
3answers
65 views
The function $f:\mathbb R^+ \rightarrow \mathbb R$ defined by $f(x)=e^{x^2/2}\int_{0}^{x}e^{-t^2/2}dt$ is
I came across the following problem:
The function $f:\mathbb R^+ \rightarrow \mathbb R$ defined by
$f(x)=e^{x^2/2}\int_{0}^{x}e^{-t^2/2}dt$ is
(A)monotone increasing,
(B)monotone decreasing,
...
4
votes
2answers
161 views
Proof that monotone functions are integrable with the classical definition of the Riemann Integral
Let $f:[a,b]\to \mathbb{R}$ be a monotone function (say stictly increasing).
Then, do for every $\epsilon>0$ exist two step functions $h,g$ so that $g\le f\le h$ and $0\le h-g\le \epsilon$?
Does ...
21
votes
7answers
669 views
Evaluate $\int_0^1\left(\frac{1}{\ln x} + \frac{1}{1-x}\right)^2 \mathrm{dx}$
Evaluate
$$\int_0^1\left(\frac{1}{\ln x} + \frac{1}{1-x}\right)^2 \mathrm{dx}$$
4
votes
1answer
245 views
Multiple Choice question about a continuous function
If $f \colon \mathbb R\to\mathbb R$ is a continuous function, which of the following statements implies that $f(0)=0$?
(A) $\int_0^1 f(x)^n \,dx\to 0$ as $n\to\infty$
(B) $\int_0^1 f\left(\frac ...
0
votes
4answers
66 views
$f:\mathbb R\rightarrow \mathbb R$ be a continuous function satisfying $f(x)=5\int_{0}^{x}f(t)dt+1$ for all $x \in \mathbb R$
I was thinking about the following problem:
Let $f:\mathbb R\rightarrow \mathbb R$ be a continuous function satisfying $f(x)=5\int_{0}^{x}f(t)dt+1$ for all $x \in \mathbb R$.Then $f(1)=?$
Please ...
11
votes
3answers
379 views
Evaluate $\int_0^1\ln(1-x)\ln x\ln(1+x) \mathrm{dx}$
Evaluate
$$\int_0^1\ln(1-x)\ln x\ln(1+x) \mathrm{dx}$$
3
votes
2answers
290 views
Prove that $\int_0^1 \psi{(x) \sin(2 n \pi x)} \space\mathrm{dx}=-\frac{\pi}{2}$
Prove that
$$\int_0^1 \psi{(x) \sin(2 n \pi x)} \space\mathrm{dx}=-\frac{\pi}{2}, \space n\ge1$$
where $\psi(x)$ - digamma function
5
votes
2answers
103 views
Prove that $\int_{0}^{1} \ln\left(\frac{1-a x}{1-a}\right) \frac{1}{\ln x} \mathrm{dx} = -\sum_{k=1}^{\infty} a^{k} \frac{\ln(1+k)}{k}, \space a<1$
Prove that
$$\int_{0}^{1} \ln\left(\frac{1-a x}{1-a}\right) \frac{1}{\ln x} \mathrm{dx} = -\sum_{k=1}^{\infty} a^{k} \frac{\ln(1+k)}{k}, \space a<1$$
I find this question rather troublesome since ...
3
votes
2answers
121 views
Prove that $\int_0^{1/e} \frac{\mathrm{dx}}{\sqrt{(\ln x)^2-1}}=K_{0}(1)$
Prove that
$$\int_0^{1/e} \frac{\mathrm{dx}}{\sqrt{(\ln x)^2-1}}=K_{0}(1)$$
where $K_{n}(x)$ is the modified bessel function of the second kind.
Some hints/suggestions?
Thanks.
6
votes
2answers
100 views
Convergence/Divergence of $\int_{0}^{1/e} \frac{\log \left(\frac{1}{x}\right)}{(\log^2 (x)-1)^{3/2}} \mathrm{dx}$
Initially I wanted to compute
$$\int_{0}^{1/e} \frac{\log \left(\frac{1}{x}\right)}{(\log^2 (x)-1)^{3/2}} \mathrm{dx}$$
but it seems that Mathematica says that the integral diverges. I thought of
...
11
votes
3answers
304 views
Compute $\lim_{n\to\infty} \left(\int_0^{\pi} \frac{\sin^2 n x}{\sin x} \ dx-\sum_{k=1}^n \frac{1}{k}\right)$
Compute the limit
$$\lim_{n\to\infty} \left(\int_0^{\pi} \frac{\sin^2 n x}{\sin x} \ dx-\sum_{k=1}^n \frac{1}{k}\right)$$
6
votes
2answers
362 views
Compute $\int_0^{\pi/2}\frac{\sin 2013x }{\sin x} \ dx\space$
How would you approach
$$\int_0^{\pi/2}\frac{\sin 2013x }{\sin x} \ dx\space?$$
The way I see here involves Dirichlet kernel. I wonder what else can we do, maybe some easy/elementary approaching ...
1
vote
5answers
123 views
Is ${\int_{0}^{+\infty}\frac{e^{-x} - e^{-2x}}{x}dx}$ finite?
I want to know if $\displaystyle{\int_{0}^{+\infty}\frac{e^{-x} - e^{-2x}}{x}dx}$ is finite, or in the other words, if the function $\displaystyle{\frac{e^{-x} - e^{-2x}}{x}}$ is integrable in the ...
