Tagged Questions

3
votes
2answers
41 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
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 ...
2
votes
3answers
90 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 ...
3
votes
6answers
142 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?
1
vote
4answers
78 views

check the convergence of the integral $\int_{0}^{\infty}\frac{1}{x\log x}\,dx$

Help me on checking the convergence of the integral $$\int_{0}^{\infty}\frac{1}{x\log x}\,dx$$ I have tried it in this way $$\int_{0}^{\infty}\frac{1}{x\log x}\,dx=\int_{0}^{\frac{1}{2}}\frac{1}{x\log ...
0
votes
2answers
69 views

check the convergence of the improper integral$\int_{0}^{1}\frac{x^{p-1}+x^{-p}}{1+x}\,dx$

How to check the convergence of the improper integral$$\int_{0}^{1}\frac{x^{p-1}+x^{-p}}{1+x}\,dx$$ I can only check that the integral is divergent for $p\geq1$, help for the cases when $p<1$. ...
0
votes
1answer
49 views

Prove convergence of improper integral using change of variable.

This may be trivial, but I could use some help... Consider a real function $f: (0,1) \rightarrow \mathbb{R}$, continuous, positive, but not necessarily bounded. Let $g: [0,1] \rightarrow [0,1]$ be a ...
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.
1
vote
1answer
90 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 ...
2
votes
2answers
68 views

Checking convergence of an improper integral

I did a quick search here but couldn't find a similar problem (it's probably out there somewhere...) I'm stuck with this rather simple improper integral: $\int_{1}^{\infty} \frac{1}{x^{\alpha}-1}dx$ ...
1
vote
4answers
95 views

Convergence of logarithm/polynomial improper integrals

My instructor has a fondness for asking questions regarding the convergence of such integrals: $$ \int_{0}^{1} \frac{\ln(x)}{x^{1/2}}\,\mathrm dx $$ $$ \int_{0}^{1} \frac{\ln(x)}{x^{3/2}}\,\mathrm dx ...
1
vote
4answers
57 views

Convergence of an improper integral - II

I'm not able to find the value of:$$ \int_a^\infty \frac{1}{x^2+1}dx, a>0 $$ What I can do?
0
votes
2answers
49 views

Convergence of an improper integral - I

What's the value of:$$ \int_a^\infty x^{-2}dx, a>0 $$ And why it converge?
2
votes
2answers
269 views

Calculating: $\lim_{n\to \infty}\int_0^\sqrt{n} {(1-\frac{x^2}{n})^n}dx$ [duplicate]

Possible Duplicate: Prove: $\lim\limits_{n \to \infty} \int_{0}^{\sqrt n}(1-\frac{x^2}{n})^ndx=\int_{0}^{\infty} e^{-x^2}dx$ I need some help calculating the above limit. What i have ...
5
votes
1answer
191 views

Does $\int_{0}^{\infty} \cos (x^2) dx$ diverge absolutely?

I believe it does, but i would like some help formulating a proof.
0
votes
0answers
83 views

Proving $\int_0^\infty\frac {1}{(1+(x\sin(5x))^2)}dx$ does not converge [duplicate]

Possible Duplicate: Why does $\int_{0}^{\infty}\frac{dx}{1+(x \sin x)^2}$ diverge? Convergence of $\int_0^\infty \frac{dx}{1+ (x^\alpha \sin x)^2}$ I understand that the following ...
6
votes
5answers
276 views

does $\int_0^\infty x/(1+x^2 \sin^2x) \mathrm dx$ converge or diverge?

$$\int_0^\infty x/(1+x^2\sin^2x) \mathrm dx$$ I'd be very happy if someone could help me out and tell me, whether the given integral converges or not (and why?). Thanks a lot.
5
votes
2answers
276 views

Does $\int_{0}^{\infty} \frac{dx}{\sqrt{x^3+x}}$ converge?

I'd like your help with checking whether $\int_{0}^{\infty} \frac{dx}{\sqrt{x^3+x}}$ converges or not. Here are the steps which led me to conclude that the integral does converge, but I'm not really ...
10
votes
2answers
313 views

Convergence/Divergence of $\int_e^\infty \frac{\sin x}{x \ln x}\;dx$

I am currently doing some project and during the course of it I need to get an answer to the following: Does $\displaystyle \int_e^\infty \frac{\sin x}{x \ln x}\;dx$ converge/ absolutely ...
1
vote
3answers
186 views

Question on convergence of improper integral

For what values of $\alpha$ is the following integral convergent? $$\int\limits_{-\infty}^{\infty}\frac{|x|^\alpha}{(1+x^2)^m}dx$$ Should the limit comparison theorem be used in this case? I am not ...
2
votes
2answers
189 views

Integrals Converging

There always seems to be a question about whether or not an integral converges. Can I ask what the best mental method is to pick the right test/process to calculate the integral? Let’s take an ...