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.
1
vote
1answer
28 views
Convergence of $\int_{0}^{1} \frac{\ln^{2} x}{x^{2}+x-2} \ dx $
How do you show that $\displaystyle \int_{0}^{1} \frac{\ln^{2}x}{x^2+x-2} \ dx $ converges?
The singularity at $x=1$ is not an issue since it is removable.
But what about at $x=0$?
4
votes
4answers
86 views
Convergence of $\int_{-\infty}^\infty \frac{1}{1+x^6}dx$
Okay, so I am asked to verify the convergence or divergence of the following improper integrals:
$$\int_{-\infty}^\infty \frac{1}{1+x^6}dx$$
and
$$\int_1^\infty \frac{x}{1-e^x}dx$$
Now, my first ...
0
votes
2answers
127 views
Prove that $\int_{-\infty}^{\infty} \sin x \, dx = 0 $
$$\int_{-\infty}^{\infty} \sin x \, dx$$
When I am doing the proof for this, why do i have to split it into
$\int_{-\infty}^a \sin x \, dx + \int_a^\infty \sin x \, dx $?
where a is a constant
1
vote
1answer
37 views
Inverse Laplace Transform. Computing the integral.
This question is related to this one, but I'm hereby taking a different approach.
Problem: Solve
$\ddot x+\delta\dot x+\omega_0^2x=\gamma\cos\omega t$.
Find the stationary points and examine their ...
1
vote
0answers
23 views
uniformly continuous and $\int_0^\infty f(t)\,\mathrm dt$ exists $\implies \lim_{x\to\infty}f(x) = 0 $ [duplicate]
I appreciate your help with this one.
Let $f \colon[0,\infty)\rightarrow \mathbb{R}$ be uniformly continuous
and let the integral $\int_0^\infty f(t)\,\mathrm dt$ exist and be final.
I need to show ...
0
votes
3answers
36 views
Evaluating $\int_0^{\infty}\frac{2}{x^2-8x+15}dx$
I am trying to evaluate $\int_0^{\infty}\frac{2}{x^2-8x+15}dx$. Factoring and using partial fraction decomposition, I have found that the indefinite integral is:
$$\ln{|x-5|}-\ln{|x-3|} + C$$
But ...
2
votes
3answers
60 views
Help evaluating $\int_0^\infty \frac{1}{x^{1/2}(x+1)}dx$
I began solving this with U sub and partial fractions...first for $x^{1/2}$ and then for $x+1$ but neither of those methods got me the answer of $\pi$.
I know the indefinite integral should be ...
2
votes
1answer
42 views
For which values of $\alpha \in \mathbb R$ two improper integrals converge
Question is: For which values of $\alpha \in \mathbb R$ the following improper integrals converge:
a.$$\int_0^1\!\left|\ln(x)\right|^\alpha\,dx$$ ...
4
votes
1answer
82 views
Integrate: $\int_0^\infty \frac{\log (1+x)}{1+x^2}dx$
Can this integral be solved with contour integral or by some application of Residue theorem?
$$\int_0^\infty \frac{\log (1+x)}{1+x^2}dx = \frac{\pi}{4}\log 2 + \text{Catlan constant}$$
It has two ...
2
votes
1answer
38 views
Proving that length of a curve is $\infty$
Let $f$ be a differentiable and continious function in $(0,1]$ and $lim_{x\to 0^+}f(x)=\infty$. Prove that the length of the curve on (0,1] is $\infty$.
Steps I tried: $L=\int_0^1 ...
5
votes
3answers
59 views
Test for convergence for improper integral $1/x^x$
I am having trouble determining if this is convergence or divergence
$$\int^1_0 1/x^x dx$$
7
votes
3answers
75 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 ...
2
votes
2answers
57 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
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
...
19
votes
2answers
176 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
152 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
53 views
Improper integral sin(x)/x converges absolutely, conditionaly or diverges?
$$\int_1^{\infty}\frac{\sin x}{x}dx$$
$$u=\frac{1}{x}$$
$$du=-\frac{1}{x^2}dx$$
$$dv=\sin xdx$$
$$v=-\cos x$$
$$\int_1^{\infty}\frac{\sin x}{x}dx=\frac{1}{x}(-\cos x)-\int_1^{\infty}\frac{\cos ...
3
votes
2answers
43 views
Convergence of $\int_0^\infty \sin(t)/t^\gamma \mathrm{d}t$
For what values of $\gamma\geq 0$ does the improper integral $$\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$$ converge? WA says it is equal to $\Gamma(4/3)$, however calculating the antiderivative seems approachless to me and can't compare ...
2
votes
1answer
73 views
$\int_0^1\frac{(f(x)-1)^2 -4x^2}{x^{3.5}}\,dx$ exists. Calculate $f(0)$ and $f'(0)$
I've tried somehow using Taylor to try and figure this one out.
Unfortunately, I couldn't seem to get a solid answer.
Thank you very much for your help!
Let f be a continuos function,
...
4
votes
2answers
95 views
Laplace transform:$\int_0^\infty \frac{\sin^4 x}{x^3} \, dx $
I have a trouble with a integral:
Using this Laplace trasform equation:
$$\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
54 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
87 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
53 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
70 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
111 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
77 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
101 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
29 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
35 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.
$$
...
3
votes
2answers
57 views
improper integral with square roots
Iv'e ran into this improper integral:
$$
\int_{0}^{1}(\sqrt{x}/\sqrt{1-x^6})dx
$$
I've tried to position $t=\sqrt x$ and i didn't get very close.
Any help will be greatly appreciated.
thank you,
...
2
votes
1answer
40 views
A parameterized elliptical integral (Legendre Elliptical Integral)
$$
K(a,\theta)=\int_{0}^{\infty}\frac{t^{-a}}{1+2t\cos(\theta)+t^{2}}dt
$$
For $$ -1<a<1;$$ $$-\pi<\theta<\pi$$
I know this integral to be a known tabulated Legendre elliptic integral, ...
5
votes
1answer
66 views
Finding a generalization for $\int_{0}^{\infty}e^{- 3\pi x^{2} }\frac{\sinh(\pi x)}{\sinh(3\pi x)}dx$
$\;\;\;\;$I was reading the introduction of Paul J. Nain's book "Dr. Euler's fabulous formula" where he talks about the sense of beauty in mathematics and quotes the G.N.Watson as saying that a ...
-1
votes
2answers
66 views
Differential Equation for improper integrals
How do I use the definiton of the improper integral to find the Laplace transform $F(s)$ for the function $f(t)=e^{(t-1)^2}$
0
votes
1answer
36 views
is f improperly integrable if g is not
$ f,g $ are nonnegative and locally integrable on $ [a,b) $ and
$ L := \lim_{x\to b-}\frac{f(x)}{g(x)}\ $ exists as extended real number.
If $ 0 < L \le \infty $ and $g$ is not improperly ...
2
votes
1answer
53 views
Improper Integral Question
Express $$\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
49 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
52 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 $$\lim_{u \to ∞} ...
2
votes
4answers
115 views
Improper Integral:$\int_{0}^{+\infty}\frac{\sin x}{x+\sin x}dx$
I want show that this improper integral convergence: $$\int_{0}^{+\infty}\frac{\sin x}{x+\sin x}dx$$ please help me.
0
votes
1answer
82 views
Problem evaluating an improper integral $\int_0^{\infty} \frac{(\sin{2x}-2x\cos{2x})^2}{x^6}$ using fourier transform
This is a question from one of the past papers of my university which I am unable to do. I am not being able to do question 2 from below.
Let $f(x)= a^2-x^2 \,\,\,\,\, |x|<a ...
0
votes
2answers
48 views
Integral $\int_{0}^{1}{\frac{\exp(-rx)}{\sqrt{1-x}}dx}$ converges for $r\geq{1}$
How do you prove that $\int_{0}^{1}{\frac{\exp(-rx)}{\sqrt{1-x}}dx}$ converges for $r\geq{1}$ ?
N.B. : I forgot everything about improper integrals, so please be very explicit :)
-2
votes
2answers
292 views
Calculate the Fourier transform of $b(x)=1/(x^2+a^2)$
I need help to calculate the Fourier transform of this funcion
$$b(x)=\frac{1}{x^{2}+a^{2}}$$
where $$a>0$$
Thanks
