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.
7
votes
5answers
175 views
Geometric Interpretation for Definite Integrals with $\pi$ in the result [duplicate]
What is the geometric interpretation for the following integral?
What is a nice geometric interpretation for the following integral (possibly in relationship to a circle) that emphasizes why we get ...
1
vote
2answers
41 views
Some more Improper Integrals
I didnt want to post all of them in one question so i split my question into 2 parts.
Basically i just can't get my head around how to prove whether or not these converge or diverge i have been thus ...
1
vote
2answers
66 views
Improper Integrals (Very Basic)
So im having a huge amount of trouble with some basic improper integrals of different types.
as they are very basic im going to post the ones i got incorrect. ( my textbook just labels converges or ...
0
votes
2answers
68 views
Improper integral of a rational function whose denominator is of degree at least two greater than that of the numerator
There's a technique in complex analysis (involving residue calculus) to solve the improper integral (from $-\infty$ to $\infty$) of a rational function whose denominator is of degree at least $2$ ...
3
votes
1answer
52 views
Convergence of this improper integral
How might I show whether $$\int_1^{\infty} \frac {x\sin x} {e^{x^3}} dx$$ converges?
Since this integrand is hard to integrate, are there standard tests (from Complex Analysis or Real Analysis) to ...
2
votes
2answers
111 views
Functions that are “only just integrable”
I am trying to find some examples of functions that are "just in $L^1([a,\infty))$" for some $a>0$ that can be chosen. By this I mean the following are integrable: $1/x^p$ and $1/x(\ln x)^p$ for ...
1
vote
3answers
78 views
How to determine whether an integral is convergent
I missed up the last lecture and can't understand how to determine whether an integral with parameters is convergent or divergent?
For example: For which values of the parameters $p,q \in ...
4
votes
1answer
78 views
Analysis on Improper Integrals
This question is from Munkres' Analysis on Manifolds, section 15 question 1.
Let $f: \mathbb{R} \to \mathbb{R}$ be the function $f(x) = x$. Show that, given $\lambda \in \mathbb{R}$, there exists a ...
6
votes
4answers
143 views
Integrating $x/(x-2)$ from $0$ to $5$
How would one go about integrating the following?
$$\int_0^5 \frac{x}{x-2} dx$$
It seems like you need to use long division, split it up into two integrals, and the use limits. I'm not quite sure ...
1
vote
1answer
42 views
Improper integrals determine if they converge or diverge.
The question is as follows.
Determine if the these 2 improper integrals converge.
$\int^{\infty}_{0} ( x^{1/2} +x^{3/2} )^{-1}$ And $\int^{\pi}_{0} (1-\cos(x))/(\sin^{2}(x))$
For the first ...
2
votes
1answer
29 views
Convergence of improper integrals and asymptotic behaviour
Is it correct to just consider the asymptotic behaviour of the integrand in an improper integral to determine whether or not it converges?
For example,
$\frac{1}{(x+3)^2}\sim_{\infty}\frac{1}{x^2}$. ...
4
votes
2answers
121 views
Calculating $\int_0^\infty(\log t)^n e^{-t}\ dt$
While messing around trying to calculate a power series for the Gamma function, I ran across this integral:
$$\int_0^\infty(\log t)^n e^{-t}\ dt,\ n \in \mathbb{N}$$
I've looked at it for a while ...
3
votes
3answers
75 views
Convergence of Improper Integrals
I am working on some exercises for Improper Integrals (not homework). The question is 1.c in Folland Advanced Calculus :
$$\int_0^\infty x^2 e^{-x^2 } \, dx$$
It asks whether the above Improper ...
12
votes
1answer
189 views
Interesting Integral $\int_{-\infty}^{\infty}\frac{e^{i nx}}{\Gamma(\alpha+x) \Gamma(\beta -x)}dx$
I am asking this question out of curiosity.
$$\int_{-\infty}^{\infty}\frac{e^{i nx}}{\Gamma(\alpha+x) \Gamma(\beta -x)}dx = \frac{ \left(2\cos \frac{n}{2} \right)^{\alpha ...
2
votes
2answers
86 views
A question about the convergence of an integral
I've a positive continuous function :
$$q : \mathbb{R} \rightarrow \mathbb{R^+}.$$
Assuming that the function $ t \mapsto t\cdot q(t) $ can be integrated (i.e.
$ \int_0^{x}{t\cdot q(t)dt}$ ...
2
votes
2answers
43 views
Proving the integral converges for all $p>1, q<1$
How can I prove that the integral
$$
\int_1^{\infty}\frac{dx}{x^p\ln^q(x)}
$$
converges when $p>1$ and $q<1$. I'm not sure where to start on this problem.
2
votes
2answers
86 views
Uniform convergence of integral on unbounded interval
Suppose I have $\;f_n:[a,\infty)\to\mathbb{R}$ and $\int_a^\infty f_n(x) dx$ exists. If $f_n\to f$ uniformly on $[a,\infty)$, I am able to show that $\int_a^\infty f(x)dx$ exists and $\int_a^\infty ...
2
votes
2answers
104 views
$\int_{-\infty}^\infty \frac{\lambda l}{2\pi\epsilon_0(x^2+l^2)^{3/2}}dx$ Proving an electric field from a wire falls off at $1/l$
If you don't know the physics behind all this, that's okay, I just need the integral of this function (or limit, I'm not too sure).
Here's the gist: normally with infinitesimal point charges, there ...
2
votes
1answer
47 views
How do I show the equivalence of the two forms of the Anderson-Darling test statistic?
It's stated in many places regarding the Anderson-Darling test statistic, which is defined as
$$n\int_{-\infty}^\infty \frac{(F_n(x) - F(x))^2}{F(x)(1 - F(x))}dF(x)$$
that this is functionally ...
1
vote
1answer
91 views
Existence of Riemann-Liouville Integral
The Riemann Liouville integral is defined as:
$\frac{1}{\Gamma\left(\nu\right)}\int\limits _{h}^{t}\left(t-\xi\right)^{\nu-1}f\left(\xi\right)d\xi$
It is supposed it does exist for all $\nu>0$ and ...
3
votes
1answer
115 views
Improper integral evaluation
I'm looking for a method to evaluate the following integral:
$\displaystyle \int_0^{\infty} \left( \frac{1}{e^x - 1} - \frac{1}{x} + \frac{e^{-x}}{2} \right) \frac{1}{x} dx$
EDIT:
Using the link, ...
5
votes
3answers
140 views
Why does $\int_{-\infty}^{\infty}e^{-x^2}\sin x\,dx=0?$
I can't get my head around something...
Why does $\displaystyle\int_{-\infty}^{\infty}e^{-x^2}\sin x\,dx=0$ but $\displaystyle\int_{-\infty}^{\infty} \sin x\,dx$ or ...
3
votes
0answers
115 views
Is there a closed form expression for this integral?
I've been trying to find a closed form expression/series expansion for the following integral without success:
$$F(a,b)=\int_{\epsilon-i\infty}^{\epsilon+i\infty} ...
4
votes
3answers
85 views
How to solve the improper integral $\int_{-\infty}^{\infty} \frac{x^2}{x^6+9}dx$ (possible trig substitution)
$$\int_{-\infty}^{\infty} \frac{x^2}{x^6+9}dx$$ I'm a bit puzzled as how to go about solving this integral. I can see that it isn't undefined on -infinity to infinity. But I just need maybe a hint on ...
2
votes
2answers
27 views
Problems with determining convergence of integral
It should be easy but I'm not sure... For which $\alpha \in \mathbb{R}$ the following integral is convergent:
$$\int_0^1 \int_0^1 \frac{1}{|y-x|^\alpha}dxdy \ \ ?$$
I get for all $\alpha \neq 1,2$ ...
5
votes
1answer
150 views
$\int_1^L\int_1^L\int_1^L\int_1^L \dfrac{\mathrm dx_1~\mathrm dx_2 ~ \mathrm dx_3 ~ \mathrm dx_4}{(x_1+x_2)(x_2+x_3)(x_3+x_4)(x_4+x_1)}$,$L\to\infty$
Let $L>1$. I am looking for the value, or the leading asymptotics for $L\to\infty$, of
$$\int_1^L\int_1^L\int_1^L\int_1^L \dfrac{\mathrm dx_1~\mathrm dx_2 ~ \mathrm dx_3 ~ \mathrm ...
3
votes
1answer
78 views
Calculating an almost Gamma integral
How would you proof that
$$I:=\int_{0}^{\infty}\frac{z^{x-1}}{e^{z}+1}dz=\left(1-2^{1-x}\right)\Gamma(x)\zeta(x)$$
I can rewrite the integral as
...
5
votes
3answers
122 views
$ \int^{\infty}_0 |\frac{1}{(1+x)\sqrt x}|^p ~ \mathrm dx < \infty \implies p=?$
If $ f(x) = \frac{1}{(1+x)\sqrt x} $ how to find all $ p > 0 $ such that
$$ \int^{\infty}_0 |f(x)|^p dx < \infty $$
The integral is with respect to lebesgue measure. Any solution or hints would ...
8
votes
2answers
155 views
Integral $\int\limits_0^\infty \prod\limits_{k=0}^\infty\frac{1+\frac{x^2}{(b+1+k)^2}}{1+\frac{x^2}{(a+k)^2}} \ dx$
Does anybody know how to prove this identity?
$$\int_0^\infty \prod_{k=0}^\infty\frac{1+\frac{x^2}{(b+1+k)^2}}{1+\frac{x^2}{(a+k)^2}} \ dx=\frac{\sqrt{\pi}}{2}\frac{\Gamma ...
2
votes
3answers
62 views
Definite Integral with a discontinuty
I have the next integral:
$$\int^{\pi/2}_0{\frac{\ln(\sin(x))}{\sqrt{x}}}dx$$
I have no clue how to start. At $x=0$ there is a clear discontinuity and I don't know how to solve the integral. The main ...
8
votes
2answers
138 views
Evaluate: $\int_{0}^\infty e^{-x^2} \cos^n(x) dx $
How to evaluate:
$$ \int_0^\infty e^{-x^2} \cos^n(x) dx$$
Someone has posted this question on fb. I hope it's not duplicate.
0
votes
0answers
98 views
Interesting integral related to floor function
Problem.
Evaluate $\displaystyle F(n, k):=\int_{0}^{1} \frac{1-\{1/x\}^n}{1-\{1/x\}^k}dx$ where $n$, $k$ are positive integers. ($\{x\}=x-\lfloor x\rfloor$)
Someone proposed this interesting problem ...
5
votes
2answers
146 views
Prove that $\int_0^{\infty} \frac{x^{2n+1}}{e^{\pi x}-1}\mathrm{d}x \in \mathbb{Q}, \forall \,n\in\mathbb{N}$
Prove that
$$\int_0^{\infty} \frac{x^{2n+1}}{e^{\pi x}-1}\mathrm{d}x \in \mathbb{Q}, \forall \,n\in\mathbb{N}$$
3
votes
1answer
115 views
Integral $\int_{-\infty}^{\infty}\frac{e^{r \arctan(ax)}+e^{-r \arctan(ax)}}{1+x^2}\cos \left( \frac{r}{2}\log(1+a^2x^2)\right)dx$
How can I show that
$$\int_{-\infty}^{\infty}\frac{e^{r \arctan(ax)}+e^{-r \arctan(ax)}}{1+x^2}\cos \left( \frac{r}{2}\log(1+a^2x^2)\right)dx = 2\pi \cos \left( r\log(1+a)\right)?$$
$a \in ...
1
vote
1answer
164 views
Integral $\int_{-\infty}^{\infty}\frac{\cos(s \arctan(ax))}{(1+x^2)(1+a^2x^2)^{s/2}}dx$
Prove that:
$$\int_{-\infty}^{\infty}\frac{\cos(s \arctan(ax))}{(1+x^2)(1+a^2x^2)^{s/2}}dx = \frac{\pi}{(1+a)^s}$$
where $a,s \in \mathbb{R}^{+}$.
This looks difficult. What would be a good start? ...
4
votes
1answer
107 views
Integration of hyperreal functions / Intermediate Value Theorem
Here's a statement on hyperreal function I've been trying to prove (I came up with it but I think it is true):
Suppose $f(x)$ is a continuous real-valued function and $h(x)$ is a continuous ...
3
votes
2answers
135 views
Why $\int_{-1}^1 \frac{1}{x^3}\, \mathrm{d} x \neq 0$?
I was considering the integral $\int_{-1}^1 \frac{1}{x^3}\, \mathrm{d} x$. At first, I suspected that it diverged due to the singularity present at $x = 0$, and WolframAlpha verified my hypothesis. ...
0
votes
0answers
51 views
Improper or Undefined
Let $f(x)=0 $ if $x\neq 1 $ and $f(1)=\infty $ then the Riemann integral
$\int_{0} ^1 f(x)$ $ dx $ = $ 0 $ or is it undefined?
If we take it as a legitimate function for improper Riemann ...
1
vote
1answer
65 views
fixing upper and lower limits
$$\int_{-\infty}^\infty \frac{e^{-x} \, dx}{1-e^{-2x}}$$
Not really sure how to fix my upper and lower limits when I get through the first substitution. Anybody know what I should do to to fix my new ...
2
votes
0answers
83 views
Integral form of $2\sum_{k=1}^{\infty}\frac{(2k-1)^2-1}{(2k-1)^4+(2k-1)^2+1}$
Being inspired by this post, I've wondered if the infinite series below may be expressed as
an intregral. I'm very curious about that.
$$2\sum_{k=1}^{\infty}\frac{(2k-1)^2-1}{(2k-1)^4+(2k-1)^2+1}$$
...
0
votes
2answers
94 views
$\int_{\frac{\pi}{2} }^{\pi} \sec x\ dx$ Converges?
$$\int_{\frac{\pi}{2} }^{\pi} \sec x\ dx$$
Does it diverges or converges?
3
votes
4answers
162 views
Integrals from MIT integration bee
$\int\frac{dx}{2+2\sin x+\cos x}$
$\int_0^{\infty}\frac{\ln x}{1+x^2}dx$
$\int\frac{dx}{x(1+x^3)}$
In general what is $\int \frac{dx}{a+b\sin x}$?
4
votes
2answers
118 views
improper integral $\int_{0}^{\infty } \frac{5x}{e^{x}-e^{-x}}$
I have a question.
Is this integral improper?
$$\int_0^\infty \frac{5x}{e^x-e^{-x}} \, dx = \int_0^a \frac{5x}{e^{x}-e^{-x}} \, dx+ \int_a^\infty \frac{5x}{e^x-e^{-x}} \, dx$$
Why is ...
6
votes
1answer
130 views
A improper integral with Glaisher-Kinkelin constant
Show that :
$$\int_0^\infty \frac{\text{e}^{-x}}{x^2} \left( \frac{1}{1-\text{e}^{-x}} - \frac{1}{x} - \frac{1}{2} \right)^2 \, \text{d}x = \frac{7}{36}-\ln A+\frac{\zeta \left( 3 \right)}{2\pi ^2}$$
...
1
vote
2answers
124 views
Calculating $\int_{-\infty}^{\infty}\frac{\sin(ax)}{x}\, dx$ using complex analysis
I am going over my complex analysis lecture notes and there is an
example about calculating $$\int_{-\infty}^{\infty}\frac{\sin(ax)}{x}\, dx$$
that I don't understand.
The solution in the notes ...
3
votes
4answers
86 views
Improper integral depending on three parameters
I am not able to prove that
$$
\int_0^\infty \frac {e^{-\alpha x}-e^{-\beta x}}{x} \sin(\gamma x)\, \mathrm{ d}x = \arctan\left( \frac {\beta}{\gamma} \right) - \arctan\left( \frac {\alpha}{\gamma} ...
3
votes
3answers
169 views
Two improper log integrals
Evaluate
$$\int_0^{\frac{\pi}{2}}\ln ^2(\tan x)\text{d}x$$
$$\int_0^{\frac{\pi}{2}}\ln ^2(\sin x)\text{d}x$$
0
votes
2answers
67 views
Exponential improper integral
How to calculate $$\int_0^\infty \frac{5x}{\theta}\left(1-\text{e}^{\frac{-x}{\theta}}\right)^4\text{e}^{\frac{-x}{\theta}}\text{d}x$$
I have done this one by expanding the terms and integrate by ...
4
votes
1answer
59 views
evaluate $\int_0^\infty \dfrac{dx}{1+x^4}$ using $\int_0^\infty \dfrac{u^{p-1}}{1+u} du$
evaluate $\int_0^\infty \dfrac{dx}{1+x^4}$using $\int_0^\infty \dfrac{u^{p-1}}{1+u} du = \dfrac{\pi}{\sin( \pi p)}$. I am having trouble finding what is $p$. I set $u = x^4$, I figure $du = 4x^3 dx$, ...
2
votes
1answer
78 views
Is there a closed form or a better solution?
This is another integral in the book "irresistible integral"
I can find that:
$$\int_0^\infty \frac{1}{\left( x^4 +2ax^2+1 \right)^{m+1}} \, \text{d}x =\frac{{{\left( -1 \right)}^m}\sqrt{2}\pi ...
