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.
5
votes
3answers
55 views
Test for convergence for improper integral $1/x^x$
I am having trouble determining if this is convergence or divergence
$$\int^1_0 1/x^x dx$$
6
votes
3answers
73 views
Integral of a rational function: Proof of $\sqrt{C}\,\int_{0}^{+\infty }{{{y^2}\over{y^2\,C+y^4-2\,y^2+1}}\;\mathrm dy}= {{\pi}\over{2}}$?
I suspect that
$$\sqrt{C}\,\int_{0}^{+\infty }{{{y^2}\over{y^2\,C+y^4-2\,y^2+1}}\;\mathrm dy}=
{{\pi}\over{2}}$$
for $C>0$.
I tried $C=1$, $C=2$, $C=42$, and $C=\frac{1}{1000}$ with Wolfram ...
1
vote
2answers
44 views
Complex-valued Fourier integral: $ \int_{ - \infty }^{ + \infty } {\frac{{\cos (ax)}}{{{x^2} + 1}}{e^{ - ibx}}\,\mathrm dx} $
I'm working on the Fourier transform, but I don't know how to evaluate the integral:
$$I = \int_{ - \infty }^{ + \infty } {\frac{{\cos (ax)}}{{{x^2} + 1}}{e^{ - ibx}}\,\mathrm dx} $$
4
votes
1answer
59 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
...
17
votes
2answers
126 views
$\int_0^{\infty}\frac{x^3}{(x^4+1)(e^x-1)}\mathrm dx$
I need to find a closed-form for the following integral. Please give me some ideas how to approach it:
$$\int_0^{\infty}\frac{x^3}{(x^4+1)(e^x-1)}\mathrm dx$$
2
votes
4answers
59 views
Approximation of alternating series $\sum_{n=1}^\infty a_n = 0.55 - (0.55)^3/3! + (0.55)^5/5! - (0.55)^7/7! + …$
$\sum_{n=1}^\infty a_n = 0.55 - (0.55)^3/3! + (0.55)^5/5! - (0.55)^7/7! + ...$
I am asked to find the no. of terms needed to approximate the partial sum to be within 0.0000001 from the convergent ...
3
votes
0answers
30 views
What is $\int_{-\infty}^{\infty} \frac{e^{-\alpha t} \cos[t + y]}{1+\beta e^{-2\alpha t} } dt$?
I want to compute the following integral:
$\int_{-\infty}^{\infty} \frac{e^{-\alpha t} \cos[t + y]}{1+\beta e^{-2\alpha t} } dt$
with $\alpha, \beta, c$ real constants, and $\alpha>0,\beta=0$.
...
15
votes
1answer
142 views
Integrating $\int^{\infty}_0 e^{-x^2}\,dx$ using Feynman's parametrization trick
I stumbled upon this short article on last weekend, it introduces an integral trick that exploits differentiation under the integral sign. On its last page, the author, Mr. Anonymous, left several ...
1
vote
1answer
44 views
Improper integral sin(x)/x converges absolutely, conditionaly or diverges?
$$\int_1^{\infty}\frac{\sin x}{x}dx$$
$$u=\frac{1}{x}$$
$$du=-\frac{1}{x^2}dx$$
$$dv=\sin xdx$$
$$v=-\cos x$$
$$\int_1^{\infty}\frac{\sin x}{x}dx=\frac{1}{x}(-\cos x)-\int_1^{\infty}\frac{\cos ...
3
votes
2answers
42 views
Convergence of $\int_0^\infty \sin(t)/t^\gamma \mathrm{d}t$
For what values of $\gamma\geq 0$ does the improper integral $$\int_0^\infty \frac{\sin(t)}{t^\gamma} \mathrm{d}t$$ converge?
In order to avoid two "critical points" $0$ and $+\infty$ I've ...
1
vote
1answer
28 views
Convergence of $\int_0^1 \sqrt[3]{\ln(1/x)} \mathrm{d}x $
Does $$\int_0^1 \sqrt[3]{\ln\left(\frac{1}{x}\right)} \mathrm{d}x$$ converge? WA says it is equal to $\Gamma(4/3)$, however calculating the antiderivative seems approachless to me and can't compare ...
2
votes
1answer
69 views
$\int_0^1\frac{(f(x)-1)^2 -4x^2}{x^{3.5}}\,dx$ exists. Calculate $f(0)$ and $f'(0)$
I've tried somehow using Taylor to try and figure this one out.
Unfortunately, I couldn't seem to get a solid answer.
Thank you very much for your help!
Let f be a continuos function,
...
4
votes
2answers
94 views
Laplace transform:$\int_0^\infty \frac{\sin^4 x}{x^3} \, dx $
I have a trouble with a integral:
Using this Laplace trasform equation:
$$\begin{align}
\int_0^\infty F(u)g(u) \, du & = \int_0^\infty f(u)G(u) \, du \\[6pt]
L[f(t)] & = F(s) \\[6pt]
...
2
votes
2answers
53 views
Improper Integral $\int_0^1\frac{dx}{x^p}$
Is this integral convergent only for $p<1$?
$$\int_0^1\frac{dx}{x^p}$$
7
votes
2answers
131 views
How can I see if this integral is convergent or not $\int_0^\infty \ \frac{1}{1 + x^4\sin x} \,dx $
I think the integral is convergent, but I don't know how to prove it.
$\int_0^\infty \ \frac{1}{1 + x^4\sin x} \,dx $
2
votes
3answers
85 views
improper integral question - $\int_{1}^{\infty}\!e^{-x}\ln x\,dx$
I ran into this integral question:
does this integral converge:
$$\int_{1}^{\infty}\!e^{-x}\ln x\,dx$$
?
Thank you very much in advance,
Yaron
3
votes
3answers
52 views
$\int_0^1 \frac{{f}(x)}{x^p} $ exists and finite $\implies f(0) = 0 $
Need some help with this question please.
Let $f$ be a continuous function and
let the improper inegral
$$\int_0^1 \frac{{f}(x)}{x^p} $$
exist and be finite for any $ p \geq 1 $.
I need to prove ...
0
votes
1answer
69 views
Improper integral $\int_{0}^{\infty}\frac{x^n}{x^{m+n+1}} \ dx=\frac{n! {(m-1)}!}{(m+n)!}.$
How can I prove that
$$\int_{0}^{\infty}\frac{x^n}{x^{m+n+1}} \ dx=\frac{n! {(m-1)}!}{(m+n)!}\quad ?$$
I tried to do induction on $n$ and on $m$, separately, but I could only do the base case ($n=1$ ...
7
votes
2answers
110 views
Proving that $\int_{0}^{\infty}\frac{\sin^{2n+1}(x)}{x} \ dx=\frac{\pi \binom{2n}{n}}{2^{2n+1}}$ by induction
I need to prove
$$\int_{0}^{\infty}\frac{\sin^{2n+1}(x)}{x} \ dx=\frac{\pi \binom{2n}{n}}{2^{2n+1}}.$$
I've seen other demonstrations of this, but they use some identities that I don't understand. ...
2
votes
2answers
84 views
How to prove that an integral doesn't exist?
$$\int_{0}^{\infty}\sin^2\left(\pi \left(x + \frac{1}{x} \right) \right) dx $$
Should I use any test for convergence?
1
vote
4answers
60 views
How can I evaluate this given improper integral?
How can I evaluate this integral:
$$\int _{ 0 }^{3}{ \frac { x }{ (3-x)^{\frac{1}{3}}} dx} \ ?$$
2
votes
4answers
75 views
Could you help me with this improper integral
How can I evaluate this improper integral?
$$\displaystyle\int_0^{\infty}\frac{1}{x(1+x^2)}\,dx $$
8
votes
1answer
94 views
Closed form for $\int_0^{\infty}\frac{\arctan x\ln(1+x^2)}{1+x^2}\sqrt{x}\,dx$
Please help me to find a closed form for this integral:
$$\int_0^{\infty}\frac{\arctan x\ln(1+x^2)}{1+x^2}\sqrt{x}\,dx$$
2
votes
2answers
32 views
What's the radius of convergence of the next sum: $\sum_{n=0}^\infty (\int_o^n\frac{\sin^2t}{\sqrt[3]{t^7+1}}dt)x^n$
What's the radius of convergence of the next sum: $$\sum_{n=0}^\infty \left(\int_0^n\frac{\sin^2t}{\sqrt[3]{t^7+1}}dt\right)x^n$$
I know that $$\int_0^\infty\frac{\sin^2t}{\sqrt[3]{t^7+1}}dt$$ does ...
1
vote
0answers
28 views
Is $f$ integrable, in the Darboux sense, on $[0,1]$?
Is
$$
f(x)=
\begin{cases}
0,\quad &x=0,\\
x\sin x,&x>0,
\end{cases}
$$
integrable, in the Darboux sense, on $[0,1]$?
I know the Darboux integral has to do with the upper sum and lower sum ...
0
votes
1answer
34 views
integral convergence - does this integration converge
i just ran into this problem and i'm having a hard time solving it:
i would like to know if this integral converges or not, and why.
i'd prefer the normal convergence tests.
$$
...
3
votes
2answers
56 views
improper integral with square roots
Iv'e ran into this improper integral:
$$
\int_{0}^{1}(\sqrt{x}/\sqrt{1-x^6})dx
$$
I've tried to position $t=\sqrt x$ and i didn't get very close.
Any help will be greatly appreciated.
thank you,
...
2
votes
1answer
35 views
A parameterized elliptical integral (Legendre Elliptical Integral)
$$
K(a,\theta)=\int_{0}^{\infty}\frac{t^{-a}}{1+2t\cos(\theta)+t^{2}}dt
$$
For $$ -1<a<1;$$ $$-\pi<\theta<\pi$$
I know this integral to be a known tabulated Legendre elliptic integral, ...
5
votes
1answer
66 views
Finding a generalization for $\int_{0}^{\infty}e^{- 3\pi x^{2} }\frac{\sinh(\pi x)}{\sinh(3\pi x)}dx$
$\;\;\;\;$I was reading the introduction of Paul J. Nain's book "Dr. Euler's fabulous formula" where he talks about the sense of beauty in mathematics and quotes the G.N.Watson as saying that a ...
-1
votes
2answers
66 views
Differential Equation for improper integrals
How do I use the definiton of the improper integral to find the Laplace transform $F(s)$ for the function $f(t)=e^{(t-1)^2}$
0
votes
1answer
35 views
is f improperly integrable if g is not
$ f,g $ are nonnegative and locally integrable on $ [a,b) $ and
$ L := \lim_{x\to b-}\frac{f(x)}{g(x)}\ $ exists as extended real number.
If $ 0 < L \le \infty $ and $g$ is not improperly ...
2
votes
1answer
53 views
Improper Integral Question
Express $$\int_0^1x^m(1-x^n)^pdx$$ in terms of gama function and hence evaluate the integral.
I used the substitution $x^n=y$ and solving got this integral as equal to the beta-function
$${1\over ...
2
votes
1answer
48 views
Improper Integrals Problem
Find the value of
$$\int_0^1(x \ln x)^3dx $$
Taking a substitute $x=e^{-y}$ i get the value as $$-3\over128$$
Does it look good ?
2
votes
2answers
54 views
$\frac{d}{dt} \int_{-\infty}^{\infty} e^{-x^2} \cos(2tx) dx$
Prove that: $\frac{d}{dt} \int_{-\infty}^{\infty} e^{-x^2} \cos(2tx) dx=\int_{-\infty}^{\infty} -2x e^{-x^2} \sin(2tx) dx$
This is my proof:
$\forall \ t \in \mathbb{R}$ (the improper integral ...
0
votes
2answers
29 views
calculate a generalized integral
I want to calculate a generalized integral:
$$\int^1_0\frac{dx}{\sqrt{1-x}}$$
I have a theorem :
if $f(x)$ is continuous over $[a,b[$ then:
$$\int^b_af(x).dx = \lim_{c\to b⁻}\int^c_af(x).dx$$
if ...
1
vote
2answers
51 views
Reinterpreting improper integrals that require Cauchy principal value to be defined
This question concerns the Cauchy principal value. Consider the improper integral $$\int_{-∞}^{∞}\frac{1+x}{1+x^2}dx$$ which is divergent, and then its Cauchy principal value $$\lim_{u \to ∞} ...
2
votes
4answers
112 views
Improper Integral:$\int_{0}^{+\infty}\frac{\sin x}{x+\sin x}dx$
I want show that this improper integral convergence: $$\int_{0}^{+\infty}\frac{\sin x}{x+\sin x}dx$$ please help me.
0
votes
1answer
79 views
Problem evaluating an improper integral $\int_0^{\infty} \frac{(\sin{2x}-2x\cos{2x})^2}{x^6}$ using fourier transform
This is a question from one of the past papers of my university which I am unable to do. I am not being able to do question 2 from below.
Let $f(x)= a^2-x^2 \,\,\,\,\, |x|<a ...
0
votes
2answers
48 views
Integral $\int_{0}^{1}{\frac{\exp(-rx)}{\sqrt{1-x}}dx}$ converges for $r\geq{1}$
How do you prove that $\int_{0}^{1}{\frac{\exp(-rx)}{\sqrt{1-x}}dx}$ converges for $r\geq{1}$ ?
N.B. : I forgot everything about improper integrals, so please be very explicit :)
-2
votes
2answers
290 views
Calculate the Fourier transform of $b(x)=1/(x^2+a^2)$
I need help to calculate the Fourier transform of this funcion
$$b(x)=\frac{1}{x^{2}+a^{2}}$$
where $$a>0$$
Thanks
1
vote
1answer
41 views
Evaluating improper integrals using laplace transform
I want to calculate the following improper integral using Laplace and transforms (and laplace transforms only).
$$\int_0^{\infty} x e^{-3x} \sin{x}\, dx$$
I propose the following method. I plan to ...
2
votes
0answers
36 views
computing a difficult integral using software
I'd like to compute the following integral. I've tried SAGE but it just runs for 15 minutes then stops.. not sure what that means. If anyone wants to take a crack with mathematica or anything, please ...
2
votes
2answers
45 views
Prove the following: Product of Roots
$1^{(1/1)} \cdot 2^{(1/2)} \cdot 3^{(1/3)} \cdot 4^{(1/4)} \cdot 5^{(1/5)} $.... diverges
well I don't really know if it does but my gut tells me it does:
I can take the log of this product
to ...
2
votes
3answers
91 views
How can I prove that $\int_1^\infty \left\lvert\frac{\sin x}{x}\right\rvert dx$ diverges?
I know a start could be to try and prove that $\int_1^\infty \frac{\sin^2x}{x} dx$ diverges since $\frac{\sin^2x}{x} \le \left\lvert\frac{\sin x}{x}\right\rvert$ in this interval, but I wouldn't know ...
0
votes
3answers
75 views
Cauchy principal value of $\int_{\infty}^{-\infty}e^{-ax^2}\cos(2abx) \,dx$
How do I find out the Cauchy Principal value of $\int_{-\infty}^{\infty}e^{-ax^2}\cos(2abx) \,dx\,\,\,\,\,\,\,\,a,b>0$ using complex integration? The answer is $\sqrt{\frac{\pi}{a}}e^{-ab^2}$, and ...
2
votes
2answers
94 views
Calculating $\int_{-\infty}^\infty e^{-ax^2}e^{ibx}dx$
In my syllabus about quantum mechanics, they state that the following integral can be easily calculated:
$$\int_{-\infty}^\infty e^{-ax^2}e^{ibx}dx = \sqrt{\frac{\pi}{a}}e^{-b^2/4a}$$
if it is ...
1
vote
2answers
66 views
Laplace transforms please help
I really need help to find to Laplace transforms of $f(x)=x+e^{-x}$, and $g(x)=xe^x$. I'm having big troubles on the calculations. Thanks.
14
votes
1answer
155 views
How to prove $\frac{\pi^2}{6}\le \int_0^{\infty} \sin(x^{\log x}) \ \mathrm dx $?
I want to prove the inequality
$$\frac{\pi^2}{6}\le \int_0^{\infty} \sin(x^{\log x}) \ \mathrm dx $$
There are some obstacles I face: the indefinite integral cannot be expressed in terms of ...
3
votes
6answers
143 views
Does the improper integral $\int_0^\infty\sin(x)\sin(x^2)\,\mathrm dx$ converge
Does the following improper integral converge?
$$\lim_{B \to \infty}\int_0^B\sin(x)\sin(x^2)\,\mathrm dx$$
2
votes
4answers
47 views
Convergence of improper integral
Show that
$\displaystyle\int_{0}^{\infty}ln(x)e^{-x}dx $
converges.
i used integration by parts but it always diverges. any hints?
