# Tagged Questions

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.

3answers
55 views

### Test for convergence for improper integral $1/x^x$

I am having trouble determining if this is convergence or divergence $$\int^1_0 1/x^x dx$$
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 ...
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: $$I = \int_{ - \infty }^{ + \infty } {\frac{{\cos (ax)}}{{{x^2} + 1}}{e^{ - ibx}}\,\mathrm dx}$$
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 ...
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: $$\int_0^{\infty}\frac{x^3}{(x^4+1)(e^x-1)}\mathrm dx$$
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 ...
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 $\alpha>0,\beta=0$. ...
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 ...
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 ...
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 ...
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 ...
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, ...
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] ...
2answers
53 views

### Improper Integral $\int_0^1\frac{dx}{x^p}$

Is this integral convergent only for $p<1$? $$\int_0^1\frac{dx}{x^p}$$
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. $\int_0^\infty \ \frac{1}{1 + x^4\sin x} \,dx$
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
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 ...
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 ($n=1$ ...
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. ...
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?
4answers
60 views

### How can I evaluate this given improper integral?

How can I evaluate this integral: $$\int _{ 0 }^{3}{ \frac { x }{ (3-x)^{\frac{1}{3}}} dx} \ ?$$
4answers
75 views

### Could you help me with this improper integral

How can I evaluate this improper integral? $$\displaystyle\int_0^{\infty}\frac{1}{x(1+x^2)}\,dx$$
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: $$\int_0^{\infty}\frac{\arctan x\ln(1+x^2)}{1+x^2}\sqrt{x}\,dx$$
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 ...
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 ...
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.  ...
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, ...
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, ...
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 ...
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 $f(t)=e^{(t-1)^2}$
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 ...
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 ...
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 ?
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 ...
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 ...
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 ∞} ...
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.
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 ...
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 :)
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
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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.
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 ...
6answers
143 views

### Does the improper integral $\int_0^\infty\sin(x)\sin(x^2)\,\mathrm dx$ converge

Does the following improper integral converge? $$\lim_{B \to \infty}\int_0^B\sin(x)\sin(x^2)\,\mathrm dx$$
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?
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