0
votes
2answers
29 views

Convergence of an improper integral(with parameters)

I'm trying to find solution to this problem: For what pairs (a; b) of positive real numbers does the improper integral $$ ...
0
votes
2answers
20 views

Show convergence of improper integral with nearest integer function

Suppose $f$ is decreasing and $\lim_{x\rightarrow\infty}f(x)=0$. Then why $$\int_0^\infty(-1)^{[x]}f(x)dx$$ converges? ($[x]$ is the nearest integer function). Any hint?
2
votes
2answers
60 views

How to see this improper integral diverges?

$$\int^\infty_1\frac{1}{x^{1+1/x}}dx$$ I'm preparing for exams. I would also like to know what are some commonly used methods to show an improper integral diverges?
2
votes
4answers
94 views

Does this improper integral converge?

$$\int^\infty_0\cos x^3dx$$ I think no, because $\cos x^3$ keeps jumping between $-1$ and $1$. How to justify this rigorously?
0
votes
1answer
22 views

Show convergence of improper integral

Suppose $f(x)>0$ and $f$ is continuous on $[0,\infty)$ and $$\lim_{x\rightarrow\infty}\frac{f(x+1)}{f(x)}<1$$ How to see that $\int^\infty_0f(x)dx$ converges? I think I should use definition. ...
0
votes
0answers
30 views

Improper integral, show if convergent/divergent.

Just want to ask a question about improper integrals. I'm a bit new to this so bare that in mind =). I have a question of the form: Calculate the following integral or show that it diverges: $$ ...
2
votes
1answer
61 views

Magical test for convergence of improper integrals?

I found this article while surfing the web. I hope it's not some kind of joke, because if it is it really fooled me. I'm trying to figure out the proof of theorem 2.3 I don't understand how the ...
9
votes
2answers
385 views

How prove this integral $\int_{0}^{\infty}f^{\alpha}(x)dx,\alpha>1$ is convergent

Question: let the function $f(x)\ge 0$,and such $$f'(x)\le\dfrac{1}{2},\forall x\ge 0$$ and this integral $\displaystyle\int_{0}^{\infty}f(x)dx$ is convergent. show that: ...
0
votes
1answer
61 views

Improper integral convergence from minus to positive infinity

Quote from Essential Calculus: Early Transcendentals, by James Stewart: If $f$ is continuous, then $$\int_{-\infty}^\infty f(x)dx=\lim_{t \to \infty}\int_{-t}^tf(x)dx$$ I thought this would be ...
1
vote
2answers
35 views

Convergent integral of divergent function

On one of my calculus lectures i was told that exist convergent inmproper integrals (in infinity) of divergent function. I was searching for an example in the internet, but I didn't find any. Has any ...
1
vote
1answer
51 views

Improper integral of Mixed Type Q

Q: Find the non-zero constant "c" such that the following integral is convergent. $$\int_{-1}^\infty \frac{e^{x/c}}{\sqrt{|x|}(x+2)}dx$$ Since the interval has both an infinite endpoint and ...
2
votes
1answer
60 views

Improper integral $\int^{1}_{0} \frac{x}{\sin{(x^{p})}} dx$

I have an improper integral with $p > 0$, $$\int^{1}_{0} \frac{x}{\sin{(x^{p})}} \ dx$$ and I want to find for which $p$ the integral exists. Now we should consider when $p = 1$ and when $p \not= ...
0
votes
2answers
74 views

Does the series from $1$ to $\infty$ of $\frac{e^{1/n}}{n^2}$ converge or diverge?

Does the series from $n = 1$ to $\infty$ of $\frac{e^{1/n}}{n^2}$ converge or diverge? Steps/tips would be greatly appreciated. Thanks!
4
votes
4answers
107 views

Again, improper integrals involving $\ln(1+x^2)$

How can I check for which values of $\alpha $ this integral $$\int_{0}^{\infty} \frac{\ln(1+x^2)}{x^\alpha}\,dx $$ converges? I managed to do this in $0$ because I know $\ln(1+x)\sim x$ near 0. I ...
6
votes
7answers
168 views

Why $ \int^{+\infty}_{-\infty} x \, dx \neq 0 $

We are going over improper integrals and tests for convergence in my Calc II course. During a lecture, my professor warned us to take caution when taking integrals from negative infinity to infinity. ...
1
vote
0answers
25 views

Testing for convergence using comparision theorem

I know that If $f(x)\geq g(x)\geq 0 \forall x$ on the interval $[a,\infty)$ then, if $f(x)$ is convergent,$g(x)$ also must converge. In doing a exercise, I have come across a problem where the use ...
1
vote
0answers
66 views

Can $\int_{1/2}^x \cot(\pi x)dx$ converge, where $x>1$.

Consider the definite integral $$\int_{1/2}^x\cot(\pi x)~dx,$$ where $x>1$. As $x\rightarrow 1$ from below $\cot(\pi x)\rightarrow-\infty$, but as $x\rightarrow 1$ from above $\cot(\pi ...
3
votes
4answers
80 views

Converging integrals

Given this integral: $$\int_2^∞ \frac {1}{({x-2)}^{2a+b}{(2x+5)}^b} \, dx$$ How do you find out when the integral converges (i.e. what limitations must be placed on a and b for the integral to ...
0
votes
1answer
34 views

Do The Integrals Have a Majorant

Consider the integrals $\int_1^\infty\frac{x^2+kx} {x^4+k^px^2+k^2}dx$. For what p do the integrands have an integrable majorant? For what p do the integrals tend to $0$? I've seen other posts like ...
1
vote
1answer
18 views

question about the convergence of a function

I'll start off with what I know: I know that if I have two functions f(x) and g(x) if: $$ \int_0^\infty f(x) < \int_0^\infty g(x) $$ If g(x) converges as the values function approaches infinity, ...
1
vote
2answers
207 views

Convergence or divergence of integral

I'm struggling with how to show that $$ \int_1^\infty \frac{x \sin x}{\sqrt{1+x^5}}dx $$ either diverges or converges. If we call the integrand $f(x)$ then $$ f(x)\leq ...
1
vote
2answers
139 views

Improper Integral Convergence of Positive Continuous Function

I ask for some help or hint how to deal with this question: Suppose f(x) is continuous and positive function for all $$x\ge a$$ Prove or provide a counterexample: If $$\int_{a}^\infty f(x)dx $$ ...
0
votes
4answers
221 views

improper integrals (comparison theorem)

In my assignment I have to evaluate the (improper) integral, by means of the "comparison theorem". And note whether the function is divergent or convergent. $$\int^{\infty}_{0} \frac{x}{x^3 + 1}dx$$ ...
0
votes
3answers
45 views

Determine if the given integral is convergent

$$\int_0^{\pi/2}{\log x\over x^a}\,\mathrm dx,\quad a<1$$ I tried solving using the $\mu-test$. so if I consider $\mu=1$ then $\lim\limits_{x\rightarrow 0} {x\log x\over x^a}$ Solving further, I ...
4
votes
5answers
684 views

Convergence of integral $\int_{0}^{\infty}{\frac{1}{x^{3}-1}}dx$

So i have the integral $$\int_{0}^{\infty}{\frac{1}{x^{3}-1}} dx$$ Software programs say that it is divergent, except for one program which evaluate numerically and gave a clear result. The ...
5
votes
2answers
145 views

Convergence of $\int_0^1 \frac{\sqrt{x-x^2}\ln(1-x)}{\sin{\pi x^2}} \mathrm{d}x.$

I would like to prove the convergence of the Newton integral $$\int_0^1 f(x)\mathrm{d}x =\int_0^1 \frac{\sqrt{x-x^2}\ln(1-x)}{\sin{\pi x^2}} \mathrm{d}x.$$ I split this into two integrals ...
3
votes
4answers
81 views

derailed in solving the convergence of $\int_{x=2}^\infty \frac{1}{xe^x} dx$ with a Taylor expansion

I'm trying to reason through whether $\int_{x=2}^\infty \frac{1}{xe^x} dx$ converges. Intuitively, It would seem that since $\int_{x=2}^\infty \frac{1}{x} dx$ diverges, then multiplying the ...
1
vote
2answers
121 views

conditional or absolute convergence of integral

i ran into a few problems where i had to check absolute\conditional convergence of a few integrals. i'm sure theres a method to check this, i just can't find the trick. i wan't help with one of the ...
0
votes
3answers
43 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 ...
4
votes
2answers
182 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
42 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
54 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
109 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
544 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
69 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
131 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 ...
1
vote
2answers
117 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
129 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 ...
1
vote
3answers
428 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 ...
1
vote
2answers
111 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
110 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
2answers
152 views

Does $\int_{1}^{\infty} \frac{\mathrm dx}{x^{\alpha}-1} $ converge?

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} ...
1
vote
4answers
195 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
69 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
59 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
373 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 ...
0
votes
1answer
106 views

speed of convergence of exponential integral

For a given $\alpha \in (0,2)$ How fast does \begin{equation} \int_{\pi/h}^\infty{\exp(-p^\alpha)}\,\mathrm{d}p \end{equation} go to zero as $h$ goes to zero? Any upper bound on the speed of ...
5
votes
1answer
213 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
117 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 ...
7
votes
5answers
394 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.