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.

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3
votes
2answers
105 views

Prove this limit $\lim \limits_{x\to\infty}f(x)=0$

I have this problem in real analysis. I think it needs integral factor or knowledge of ODE to prove, but not sure how to it. Here is the question: Let $f$ be a real valued continuous function on ...
5
votes
2answers
142 views

Prove or disprove $\int_{-\infty}^\infty \frac{dx}{\cos x+\cosh x}=\frac{1512835691 \pi}{1983703776}$

In this question, Evaluating the integral $\int_{-\infty}^\infty \frac {dx}{\cos x + \cosh x}$ , robjohn evaluates the integral to a nice summation with an approximate value. When plugged into W|A, it ...
0
votes
1answer
14 views

Improper integral confusing step

The following passage is in my textbook: $$A(S) = \int_0^{\infty} f(E) \max(S-E,0)dE$$ This simplifies to $$A(S) = \int_0^{S} f(E)(S-E) dE$$ Now this is from a finance textbook so it might ...
0
votes
0answers
17 views

Principal value methods for fourier laplace etc.

I recently saw this here: $$\int_{-\infty}^{\infty} \frac{1}{\omega^2} e^{j\omega t} d\omega$$ and I was unable to understand how such an integral could be computed. I want to learn about this ...
1
vote
2answers
10 views

Discretization of integral on infinite domain.

Let $[a, b]$ be a closed interval of the real line and let a sequence as $$a = x_0 \le t_1 \le x_1 \le t_2 \le x_2 \le \cdots \le x_{n-1} \le t_n \le x_n = b . \,\!$$ This partitions the interval $[a, ...
1
vote
1answer
102 views

Integral of $xe^{-ax^2-bx^{-1}}$

I am currently facing an integral I have no clue how to solve it. I believe it is rather exoctic, but I hope you might have some good advice: $$\int_0^{\infty} x e^{-ax^2-bx^{-1}} \, \mathrm{d}x$$ ...
2
votes
3answers
57 views

Evaluating $\int_{-\infty}^{\infty}e^{-x^2}dx$ using polar coordinates. [duplicate]

I need to express the following improper integral as a double integral of $x$ and $y$ and then, using polar coordinates, evaluate it. $$I=\int_{-\infty}^{\infty}e^{-x^2}dx$$ Plotting it, we find a ...
0
votes
3answers
96 views

difficult limit with a improper integral

It is assigned at my calculus class the following problem. problem: Evaluate the following limit $$\displaystyle \lim_{n \to \infty} \int \limits_{\frac{1}{(n+1)^2}}^{\frac{1}{n^2}} ...
5
votes
3answers
304 views

Decomposition into partial fractions to compute an integral

I'm having problems with: $$\int_{-\infty}^{\infty}\frac{x^4+1}{x^6+1}dx$$ I was thinking: $\frac{x^4+1}{x^6+1}$ is an even function and the interval $(-\infty,\infty)$ is symmetric about 0, we ...
2
votes
3answers
48 views

Improper Integral: $\int_{-\infty}^\infty\frac{e^{-t}}{1+e^{-2t}}\ dt$

$$\int_{-\infty}^\infty\frac{e^{-t}}{1+e^{-2t}}\ dt$$ I have the antiderivative as $$-\arctan e^{-t}$$ but when I do it out, I end up getting $$-\frac\pi4 + 0 - \frac\pi2+\frac\pi4$$ However, I ...
0
votes
1answer
27 views

Parametric integral and equivalence in $\infty$

I have to find a equivalent when $x$ comes to $\infty$ for all $a$ (fixed) in $\mathbb{R}_+^*$ of this integral : $$ \int_0^a \frac{e^{-xt}}{\sqrt{a-t}}\mathrm{d}t $$ My work : For $x \in ...
2
votes
1answer
59 views

A real integral (may be requires contour integration)?

The integral I have in mind is $$\int^\infty_0 x^{r}(x + \lambda)^{-1}dx$$ where $r \in (-1, 0)$, and $\lambda$ is a non-negative constant. I apologize if this is really easy and I am missing some ...
1
vote
2answers
58 views

Find functions $f$ and $\alpha$ such that $\int_1^{\infty}|f|d\alpha$ converges, but $\int_1^{\infty}fd\alpha$ does not exist?

Find functions $f$ and $\alpha$ such that the improper Riemann-Stieltjes integral $\int_1^{\infty}|f|d\alpha$ converges, but $\int_1^{\infty}fd\alpha$ does not exist? I'm really not sure how to start ...
1
vote
1answer
38 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
65 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 ...
2
votes
2answers
52 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 ...
0
votes
1answer
65 views

Exponential integral of sine

How can I calculate the following integral: $$ \int_{-\infty }^{\infty} e^{-x^{2} + sin x}dx$$ Thank you very much!
1
vote
0answers
81 views

Evaluate $\lim_{n\to\infty}\int_0^{\infty}\cos^n(x)dx$ [closed]

How can I solve that $\lim_{n\to\infty}\int_0^{\infty}\cos^n(x)dx$?
1
vote
2answers
32 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?
3
votes
2answers
98 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 $x=\frac{1-t}{t+1}$ ...
1
vote
2answers
58 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) = - ...
9
votes
3answers
201 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 ...
1
vote
1answer
38 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 ...
7
votes
0answers
46 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 ...
6
votes
0answers
159 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
37 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 ...
0
votes
1answer
22 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 ...
1
vote
0answers
31 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 ...
0
votes
1answer
44 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
1answer
58 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 ...
1
vote
1answer
55 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 ...
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 ...
-2
votes
1answer
31 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
1answer
19 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
2answers
72 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 ...
2
votes
2answers
28 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} \! ...
1
vote
1answer
27 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: $$F(x)=\int_0^x f(t) dt\sim 0 + x f(0), \ \ \ (x\rightarrow 0)$$ ...
5
votes
2answers
103 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 ...
0
votes
2answers
36 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
47 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
67 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 ...
0
votes
1answer
81 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 ...
4
votes
4answers
263 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 ...
0
votes
2answers
43 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 ...
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
2answers
71 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 ...
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 ...
0
votes
1answer
41 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.
0
votes
2answers
59 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
29 views

Sum of integrals with variables shifted in each sum: How to justify this expression?

I annoyingly can't justify a step in the solution of the following problem. I have the following expression at hand: $$ \sum_{n=1}^{N}\int_{-\infty}^{\infty}{(y(x_n + \xi) - t_n})\nu(\xi)\eta(x_n + ...