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.

learn more… | top users | synonyms

1
vote
1answer
35 views

Let $a_n>0$ for $n \geq 1$ and let series: $\sum_{n=1}^{\infty}a_n$ diverge. Let $S_n=a_1+a_2+…+a_n > 1$ for $n \geq 1$

Prove that the series: $$\sum_{n=1}^{\infty}\frac{a_{n+1}}{S_n \ln S_n}$$ diverges and the series : $$\sum_{n=1}^{\infty}\frac{a_{n}}{S_n \ln^2 S_n}$$ converges. (Using the famous criteria I ...
1
vote
1answer
22 views

For what $\alpha$ does the integral absolutely and for what conditionally converge?

For what $\alpha$ does the integral absolutely and for what conditionally converge ? $$\int_{0}^{1}\frac{\ln^{\alpha} (1+x^4)}{x^4}\cos{1 \over x}dx$$ Not sure which criteria to use to prove the ...
1
vote
2answers
56 views

Find $f$ so that $\int_{1}^{\infty}f(x)dx$ exists, but $\displaystyle \lim_{x\to\infty}f(x)$ does not exist? [duplicate]

I need help finding an example of a function such that $\int_{1}^{\infty}f(x)dx$ converges, but $\displaystyle \lim_{x\to\infty}f(x)$ does not exist. I was trying to find examples of functions ...
1
vote
2answers
48 views

Given $|f(x)|=1$,how to construct an $f(x)$, such that $\int ^{+\infty }_{0}f\left( x\right) dx$ converges

Here's the problem: Given $|f(x)| = 1$, construct an $f(x)$, such that $$\int ^{+\infty }_{0}f\left( x\right) dx$$ converges. I think this problem may be done by dividing the 1s and -1s smartly, but ...
-1
votes
1answer
21 views

Gamma and Beta Functions [closed]

\begin{equation*} \int \limits _0 ^\infty x^m \mathbb e ^{-x^n} \mathbb d x = \frac 1 n \Gamma (\frac {m+1} n), \space m>-1, \space n>0. \end{equation*} \begin{equation*} \int \limits _0 ^1 ...
0
votes
2answers
42 views

Consider the intergal $I=\int_{1}^{\infty}e^{ax^2+bx+c}dx$, where $a,b,c$ are constants. When does the integral converge? [closed]

Consider the integral $I=\displaystyle\int_{1}^{\infty}e^{ax^2+bx+c}dx$, where $a,b,c$ are constants. When does the integral converge? As usual, these are alien concepts to me, it gets tough to ...
2
votes
0answers
78 views
1
vote
2answers
30 views

Convergence of a double integral

Is the integral $$\int_1^\infty\int_{e^{-x}}^1\frac{\sin y}{x^2y}dy dx$$ convergent or divergent?
1
vote
1answer
35 views

Definite integral including natural log, cosine, and hyperbolic sine

Here is an integral question I have, I am solving some other problems like this but I am stumped on this one: $$\int_0^{\pi+1}\frac {\ln(\cos(x+1))}{\sinh(x^2)}dx$$ I used some methods such as ...
3
votes
2answers
67 views

uniform bound for sine integral function

Prove that for any $0<a<b$, $$ \left|\int_a^b\frac{\sin x}{x}\,dx\right|\le4 $$ Here is my approach. I used integration by parts to prove that LHS is bounded by $3$ when $a\ge 1$. I will be done ...
7
votes
3answers
170 views

How to compute $\int_0^\infty \frac{x^4}{(x^4+ x^2 +1)^3} dx =\frac{\pi}{48\sqrt{3}}$?

$$\int_0^\infty \frac{x^4}{(x^4+ x^2 +1)^3} dx =\frac{\pi}{48\sqrt{3}}$$ I have difficulty to evaluating above integrals. First i try the subsititue $x^4 =t$ or $x^4 +x^2+1 =t$ but it makes ...
3
votes
2answers
90 views

How to evaluate $\int_0^1 \ln(\frac{1+x}{1-x}) \frac{dx}{x} = \frac{\pi^2}{4}$?

Can anyone suggest the method of computing $\int_0^1 \ln(\frac{1+x}{1-x}) \frac{dx}{x} = \frac{\pi^2}{4}$ ? My trial is following first set $t =\frac{1-x}{1+x}$ which gives $x=\frac{1-t}{t+1}$ ...
3
votes
1answer
182 views

Analyzing the convergence of an improper integral

I have to analyze the convergence of $$\int _{0}^{+\infty} \frac{\cos \left( x\right) -1} {x^{5 / 2}+5x^{3}}\,dx$$ I've rewritten the integral as $$ \int _{0}^{+\infty} \frac{\cos \left( x\right) ...
1
vote
2answers
53 views

How to compute $\int_0^1 \frac{x-1}{\ln(x)} dx = \ln(2)$? and $\int_0^\infty \ln(t) e^{-t} dt $?

$\int_0^1 \frac{x-1}{\ln(x)} dx = \ln(2)$ First i try $\ln(x)=t$ so that $\frac{1}{x} dx =dt$ then integral becomes \begin{align} &\int_{-\infty}^{0}\frac{e^t-1}{t} (e^t dt) = - ...
85
votes
19answers
19k views

Solving the integral $\int_{0}^{\infty} \frac{\sin{x}}{x} \ dx = \frac{\pi}{2}$?

A famous exercise which one encounters while doing Complex Analysis (Residue theory) is to prove that the given integral: $$\int_{0}^{\infty} \frac{\sin{x}}{x} \, dx = \frac{\pi}{2}$$ Well, can ...
6
votes
0answers
31 views

Does such divergent integral assume the same values for any regularization?

Consider the integral: $$\int_0^\infty\sin(x)dx.\tag1$$ It's clearly divergent, but if we regularize it as $$\int_0^\infty\sin(x)e^{-x/a}dx=\frac{a^2}{a^2+1},\tag2$$ we can take the limit of ...
1
vote
2answers
95 views

Closed expression of the following integral?

I believe that the following integral has a closed expression, but I haven't been able to check it $$I(k)=\int_{-\infty}^{\infty}dt\,\text{erf}\left(\frac{t}{b}-i \frac{1}{2}b(k+a)\right) ...
0
votes
1answer
19 views

convergence of $\int_{x=0}^{\infty}x^{-\frac{M-1}{2}-N}(1-e^{-x})^{M-1}e^{-x}dx$

I am trying to study the convergence of $$\int_{x=0}^{\infty}x^{-\frac{M-1}{2}-N}(1-e^{-x})^{M-1}e^{-x}dx,$$ where $M$ and $N$ are positive integers. I've tried some $M$ and $N$, and it seems that ...
4
votes
0answers
68 views

How to find the value of this integral?

This integral to the value \begin{align} \int_0^1\frac{\ln^2(1+x)\ln^2 x}{1-x}\ dx=&\ ...
0
votes
2answers
33 views

Proving Integral Test?

Assume that $f(x) \geq 0$ and that $f$ decreases monotonically on $[1, \infty]$. Prove $\int_{1}^{\infty} f(x)dx$ converges iff $\sum_{n=1}^{\infty} f(n)$ converges. My proof: If $f$ is non-negative ...
4
votes
2answers
941 views

Definition of the gamma function

I know that the Gamma function with argument $(-\frac{1}{ 2})$ -- in other words $\Gamma(-\frac{1}{2})$ is equal to $-2\pi^{1/2}$. However, the definition of $\Gamma(k)=\int_0^\infty t^{k-1}e^{-t}dt$ ...
1
vote
3answers
245 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
34 views

Improper integral and lower Riemann sums

Given $f$ is positive and continuous on $(0,1]$ and its improper integral exists there. Is it true that the lower Riemann sums converges to the integral? I'm thinking about using definition but reach ...
0
votes
1answer
42 views

Computing $\int_{0}^1 \frac{\left( (1-x )a +x b\right)^2}{(1-x)c +x d} dx$

I want to find the integral of \begin{align*} \int_{0}^1 \frac{\left( (1-x )a +x b\right)^2}{(1-x)c +x d}dx \end{align*} for any $a,b$ and $c>0$ and $d>0$. Using Wolfram-Alpha I found that ...
1
vote
0answers
26 views

The Laplace transform of $\exp(t^2)$

A naive attempt to calculate the Laplace transform of the function $f(t)=e^{t^2}$ results in integrals of the form $$\int_0^\infty e^{t^2-st}dt,$$ which obviously don't exist as the integrand grows ...
5
votes
2answers
101 views

How to compute $\int_{-1}^{1} e^{-1/(1-x^2)}dx$?

As in the title, I would like to compute the integral: \begin{equation} \int_{-1}^{1}e^{-1/(1-x^2)}dx \end{equation} My hunch tells me that I should try to transform it to the correspoding ...
1
vote
1answer
71 views

Integral of $\int_{-\infty}^\infty\frac{e^{-y^2/2} \sinh(cy)^2} {\cosh(Mcy) }dy$

Is it difficult to compute or find a good computable lower bound on the integral \begin{align*} \int_{-\infty}^\infty\frac{e^{-y^2/2} \sinh(cy)^2} {\cosh(Mcy) }dy \end{align*} where $c$ and $M$ are ...
1
vote
1answer
47 views

Gamma and Beta function proof.

I'm trying to proof the equality $B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$ when $x,y>0,$ without using calculus in many variables. I've investigated about the topic but all references make ...
1
vote
1answer
54 views

Evaluate $\int_{x=0}^{\infty}\left(\frac{1}{\sqrt{x}}(1-e^{-x})\right)^{M-1}e^{-x}(1+sx)^{-N}dx,$

I am trying to evaluate $$\int_{x=0}^{\infty}\left(\frac{1}{\sqrt{x}}(1-e^{-x})\right)^{M-1}e^{-x}(1+sx)^{-N}dx,$$ where $s>0$, $M$ and $N$ are positive integers. But seem that the above integral ...
0
votes
2answers
34 views

When using the Integral test, why is the value of the integral different from the sum of the series?

According to my textbook, the value of the improper integral is not always equal to the sum of the series. But why is that?
0
votes
1answer
44 views
0
votes
1answer
17 views

Infinite Integral of a Bessel Function

I need to calculate the following integral $$ \int_0^{\infty}xdxJ_n(kx) $$ Integrating it by parts and using the normalization of Bessel functions, I find it (somewhat heuristically) to equal the ...
4
votes
1answer
58 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
1answer
23 views

Proving an integral identity

I'm dealing with the Hermitian operator, and I've been asked to prove that all $f(x) = x^n e^{\alpha x}$ belong to $L^2(-\infty,\infty;e^{-x^2/2})$ by showing that: $$\int_{-\infty}^{\infty}x^m ...
3
votes
1answer
48 views

Convergence of a integral

The question is: exists a natural number $n \geq 2$such that $$ \displaystyle\int_{0}^{+ \infty} \displaystyle\frac{\ln r}{(1 + r^2)^{n}} r dr< \infty ?$$ I am trying to do this : i know that ...
4
votes
2answers
68 views

Show that $ \int_{-\infty}^{\infty} \frac{x^3}{(x^2+4)(x^2+1)}\, dx$ does not converge

I noticed that $\displaystyle \int_{-a}^{b} \frac{x^3}{(x^2+4)(x^2+1)}$ will converge to $0$ whenever $a=b$ and will converge to some value whenever $a,b$ are in the reals (excluding infinity). How ...
15
votes
2answers
337 views

Improper Integral $\int_0^\frac{1}{2}x^n\cot(\pi x)\,dx$

What is the closed form of the following integral for every $n\in\mathbb{N}$? $$\int_0^\frac{1}{2}x^n\cot(\pi x)\,dx$$ By Mathematica we see that $$\int_0^\frac{1}{2}x\cot(\pi ...
1
vote
1answer
22 views

First order approximation of $F(x)=\int_0^x f(t) dt$ in the neighbourhood of $\infty$

Let $f(x)$ continuous on the real line. Then the first order approximation of $$F(x)=\int_0^x f(t) dt$$ in the neighbourhood of $0$ is: $$F(x)=\int_0^x f(t) dt\sim 0 + x f(0), \ \ \ (x\rightarrow 0)$$ ...
2
votes
2answers
26 views

explain the solution and/or suggest a different one

I have come across the following problem, in my calculus II course, about improper integrals: problem: Find the following limit, if it exists. $\displaystyle\lim_{x\to 1} \int\limits_{x}^{x^2} \! ...
0
votes
1answer
45 views

Proving $\int_0^\infty e^{-ax}x^n\,dx = \frac{1}{a^{n+1}} \Gamma(n+1)$

Prove that $$ \int_{0}^{\infty} \ e^ {-ax} x^{n} dx = \frac{1}{a^{n+1}} \Gamma(n+1) \qquad (n>-1, \, a>0). $$ My try: Let $dv = e^{-ax}$ and $u = x^n$. Then $v = -\frac{1}{a}e^{-ax}$ ...
4
votes
3answers
63 views

The shortest way to prove that $\int_1^\infty \frac{{\arctan \left( x \right)}}{{\sqrt {{x^4} - 1} }}dx $ converges.

I'm trying to show that the integral $$\int_1^\infty \frac{{\arctan \left( x \right)}}{{\sqrt {{x^4} - 1} }}dx \quad \text{is convergent}.$$ We know that $$\frac{{\arctan \left( x \right)}}{{\sqrt ...
13
votes
2answers
200 views

Closed form $\int_{-1}^{1} \frac{\ln (\sqrt{3} x +2)}{\sqrt{1-x^{2}} (\sqrt{3} x + 2)^{n}}\ dx$

Does the following integral $$\int_{-1}^{1} \frac{\ln (\sqrt{3} x +2)}{\sqrt{1-x^{2}} (\sqrt{3} x + 2)^{n}}\ dx, \; \; n \in \mathbb{N}$$ have a nice closed form? Basically I cannot tackle it in any ...
3
votes
2answers
115 views

The integral of $e^{-x^2}$ [duplicate]

How can I integrate this by parts? It seems to become recursive. I'm familiar with the classical solution, and cannot use that here due to the constraints of this class. Here's the integral (to ...
4
votes
4answers
254 views

Can we determinine the convergence of $\int_0^\infty \frac{x^{2n - 1}}{(x^2 + 1)^{n + 3}}\,dx$ without evaluating it?

Can we determine convergence without evaluating this improper integral? $$\int_0^\infty {\frac{x^{2n - 1}}{{\left( x^2 + 1 \right)}^{n + 3}}\,dx}\quad\quad n\geq 1\;,\; n\in\mathbb{Z}$$ When ...
2
votes
2answers
68 views

Manipulating $\int_{x_{0}}^{\infty} \frac{1}{x} \, \cos (x t) \, \text{e}^{-x^{2}} \, dx$

Is there a way to express the integral $I(x_{0}, t) = \int_{x_{0}}^{\infty} \frac{1}{x} \, \cos (x t) \, \text{e}^{-x^{2}} \, dx$, where $x_{0} \neq 0$ and $t \ge 0$, in terms of more well-known ...
3
votes
1answer
56 views

Computing $\int_{0}^{\pi/2}\cos(x)\ln(\tan(x))dx$

Compute $\int_{0}^{\pi/2}\cos(x)\ln(\tan(x))dx$ It is easy to check this improper integral converges. One also notes that ...
2
votes
3answers
74 views

How evaluate $\int \frac{\cos^2(x)}{1 + \text{e}^x}dx$ to find an improper integral

Can someone help me evaluate this: $$\int \frac{\cos^2(x)}{1 + \text{e}^x}dx\;?$$ I need it for determining whether the improper integral $\int_0^\infty {\frac{{\cos^2{{(x)}}}}{{1 + ...
17
votes
6answers
635 views

Improper Integral $\int_{0}^{1/2}\left(2x - 1\right)^6\log^2\left(2\sin\pi x\right)\,dx$

How can I find a closed form for the following integral $$\int_0^{1/2}\left(2x - 1\right)^6\log^2\left(2\sin\pi x\right) \,dx$$
0
votes
2answers
54 views

Improper integral problem.

How to find divergence/convergence condition for $p$ on $$\int\limits_{2}^{\infty} \frac{1}{{(\ln x)}^p} \, \mathrm d x$$ I tried comparison test , but failed.
0
votes
1answer
39 views

For what values is this integral convergent?

How can I find for what values of $r$ $$\int_0^\infty x^re^{-x}dx$$ converges? I started by rewriting it as $$\lim_{b\to\infty}\int_0^bx^re^{-x}dx$$ but am not sure how to figure it out from here.