1
vote
1answer
22 views

Convergence of an Improper Integral $\int_{-\infty}^{\infty}\cos(x\log\left|x\right|)dx$

This is a question from an old exam qualifier: Show that the improper integral $\int_{-\infty}^{\infty}\cos(x\log\left|x\right|)dx$ is convergent. I first notice that \begin{equation*} ...
0
votes
1answer
16 views

Showing the convergence of improper integral.

Hello I have to show that this improper integral is convergent: $$\int_{0}^{1} \frac{e^{\frac{-1}{x}}}{x^2} dx$$ , but I don't have any starting ideea. Any tips would be great, thank you.
1
vote
2answers
71 views

Convergence of $\displaystyle\int\frac{1}{\sqrt[3]{1-x^3}}\ dx$

Please help me to prove that this integral converges. $$\int_{0}^1 \frac{1}{\sqrt[3]{1-x^3}}\ dx $$ No ideas. Tried to find function which is bigger and converges, equivalent fun-s, but no result ...
-1
votes
2answers
31 views

A simple convergent integral but not absolutely convergent.

Anybody knows a simple example for convergent function but not absolutely convergent? ( simple = easy ) Thanks for coments!!!
0
votes
1answer
62 views

How to prove that integral of function is convergent

$\int_{0}^{\infty} \frac{(\sin(x) )}{ x} \,\mathrm dx$ and $\int_{0}^{\infty} \frac{(\sin(x) \arctan(x))}{ x} \,\mathrm dx$ These are convergent. How to prove that?? I using the comparison test. ...
3
votes
2answers
88 views

How to calculate the improper integral $\int_0^\infty\left(\frac{1}{\sqrt{x^2+4}}-\frac{P}{x+2}\right)dx$

This is the first time I've seen a problem like this. I have no idea what to do. Detailed guidance would be of great help. For which values of P does the integral converge? ...
2
votes
0answers
50 views

Multivariable Integral Calculus help

I have two questions. First: Is my proof "strong" enough? I am being asked to prove that $$\int_{0}^\infty\int_{0}^x e^{-sx}f(x-y,y) dydx = \int_{0}^\infty\int_{0}^\infty e^{-s(u+v)}f(u,v) dudv$$ ...
0
votes
2answers
38 views

Show that Improper integral converges

Show that the following integral is convergent. $$ \int_{1}^{\infty}\sin\left(1 \over x^{2}\right)\cos\left(x^{2}\right)\,{\rm d}x $$ Not sure how I can solve ...
2
votes
2answers
50 views

Convergence of $\int_{-\infty}^{\infty}f(x)dx$

I posed a question to my calculus professor, asking how to evaluate the Riemann integral $$\int_{-\infty}^\infty f(x) \, dx$$ I can simplify the above integral as $$\int_{-\infty}^{\infty}f(x)\,dx ...
0
votes
0answers
37 views

For what $p$, $\int_0^{\infty} \frac{\ln (1+x^{3p})}{(x+x^3)^{4p}arctg\sqrt{x}}$ converges

I have to see for what values of $p$ the following integral converges. $$\int_0^{\infty} \frac{\ln (1+x^{3p})}{(x+x^3)^{4p}arctg\sqrt{x}}=\int_0^1 \frac{\ln (1+x^{3p})}{(x+x^3)^{4p}arctg\sqrt{x}} + ...
0
votes
0answers
20 views

Convergent Improper Integral help

I am currently studying improper integrals and came across the following problem. Analyze the convergence of the improper integral of $f(x,y) = 1 / ( x^4 + y ^2 ) $ over $R = \{(x,y) : x\geq 1, y\geq ...
4
votes
1answer
43 views

Does the integral $\int_0^1\frac{\sin x}{\sqrt{x^3}}\cos\left(\frac1x\right)dx$ converge?

I tried to prove convergence this way: Suppose: $f(x)=\left|\frac{\sin x\cos\left(\frac1x\right)}{\sqrt{x^3}}\right|$, $g(x)=\left|\frac{\cos\frac1x}{\sqrt x}\right|$. $\lim_{x\to ...
0
votes
1answer
23 views

A question about convergence of improper parametric integral

Could you give me some hint how to find all $\alpha\in R$ for with the integral $\int_0^1 \frac{a-x^{\alpha}}{1-x}$ converges. Is clear that this integral converges for all$\alpha\in N$, but I could ...
1
vote
2answers
40 views

Discuss the convergence of $\int_0^1x^n \left[\log({1\over x})\right]^m \, dx$

Discuss the convergence of $$\int_0^1x^n\left[\log\left({1\over x}\right)\right]^m \, dx$$ Need some clues. I know that both $0$ and $1$ are points of discontinuities.
0
votes
3answers
56 views

Determining the convergence of $\int_{1}^{2}\frac{1}{\sqrt{x}(2-x)}\, \mathrm{d}x$ in simple way.

Is there a (simple) way to determine its convergence without determining its value? I know that the function $x\mapsto \left (\sqrt{x}(2-x) \right )^{-1}$ is continuous for all $x\in ...
3
votes
1answer
54 views

When $\int_{0}^{\infty}f(x)dx=\sum_{n=0}^{\infty}\int_{n}^{n+1}f(x)dx$?

Is the following always true? (i.e. if both converges, limits are equal; if one diverges, the other must diverge; EXCLUDE the case where the limit keeps "jumping") $$ ...
2
votes
0answers
33 views

Question about convergence of improper integral

Could you give me some hint how to solve this problem: Suppose $f$ is continuous on $(0,1]$ and there is $M$ such as $\left|\int_x^1f(t)\, dt \right|\le M$. Prove that $\int_0^1f(x)\, dx$ converges ...
1
vote
0answers
56 views

How to prove convergence of $\int_0^1f\left(\sqrt x \right)dx$?

Could you please give me some hint how to prove convergence of $\int_0^1f\left(\sqrt x \right)dx$ when f(x) is continuous for $0<x\le1$ and $\lim_{x\to0^+}x^3f^2(x)=1$ ? I tried the usual way: ...
0
votes
1answer
47 views

A question about improper integral

Could you please give me some hint how to solve this problem: Suppose f(x) continuous in $[0,\infty)$ and for each a,b>0 and c>b $ab \left|\int_0^1 f\left(ax+c \right) dx \right|<1$. Prove ...
0
votes
2answers
43 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
27 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
66 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
124 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
36 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
43 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
65 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
413 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
86 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
55 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
66 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
82 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
116 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
184 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
26 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
70 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 ...
1
vote
1answer
19 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
239 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
170 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
298 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
47 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
790 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
151 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
82 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
146 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
219 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
45 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 ...