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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1answer
80 views

How to get a result in Integration

$$I(a)=\displaystyle\int_{0}^{1} \frac{tan^{-1}ax}{x\sqrt{1-x^{2}}} dx$$ By using Leibniz's formula, $$I'(a)=\displaystyle\int_{0}^{1} \frac{\partial}{\partial a} \frac{tan^{-1}ax}{x\sqrt{1-x^{2}}} ...
4
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0answers
90 views

Double integral of symmetric polylogarithmic function over rectangular region

This question was inspired by M.N.C.E.'s wonderful response here. While exploring the possibility of generalizing his result, I found that a significant part of the problem reduced to evaluating the ...
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1answer
66 views

Determine whether a function is improperly Riemann Integrable?

My attempt: i) $$\lim_{t\to\ 0} \int_t^1 \ x^{-0.5}\ $$ = $$\lim_{t\to\ 0} 2(1^{0.5}) - 2(t^{0.5}) = 2 - 0 = 2 $$ (since $lim_{t\to\ 0^+} t^{0.5} = lim_{t\to\ 0^-} t^{0.5} = 0 $) ii) $$\lim_{t\to\ ...
5
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1answer
146 views

How to compute $\int_{2/\pi}^{+\infty }\ln\cos(1/x)\,dx$? [closed]

What it says in the title. If $$I=\int_{2/\pi}^{+\infty }\ln\left({\cos\left({\frac{1}{x} }\right) }\right) \, \mathrm dx,$$ how could I proceed in order to find the value of $I$?
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1answer
49 views

Integration by Parts Problem: Help in understanding why a part of it equals 0

$$4I= \int_0^{\infty} \frac{4x^3 +\sin(3x)-3\sin x}{x^5} \ \mathrm{d}x $$ $$=\frac{-1}{4} \underbrace{\left[\frac{4x^3+\sin(3x)- 3 \sin x}{x^4} \right]_0^{\infty}}_{=0} +\frac{1}{4} \int_0^{\infty} \...
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1answer
34 views

Proving an equation involving integrals and limits

I have to show the following equation: $\large\int_0^\infty \! e^{-st}\cos(\beta t) \, \mathrm{d}t=\frac{s}{s^2+\beta ^2}$ with $s>0$ I've come so far: $\large\int_0^\infty \! e^{-st}\cos(\beta ...
4
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3answers
127 views

Problem with Differentiation under the Integral Sign

Problem: Evaluate: $$\displaystyle\int_{0}^{\infty} \dfrac{1}{x} \left(\tan^{-1}(\pi x) - \tan^{-1}x\right)dx.$$ My incorrect attempt: $$\displaystyle\int_{0}^{\infty} \dfrac{1}{x} \left(\tan^{-...
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2answers
36 views

How to formalize my argument for existence of $\lim \limits_{\epsilon \to 0^{+}} \int_{\epsilon}^{c} \frac{\sin x}{x} dx$

I know this exists - $\frac{\sin x}{x}$ is not defined on $0$ but it has an asymptote there. However, I'm finding it difficult to formalize my argument in a way that I don't say things which might not ...
3
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2answers
67 views

Help with Differentiating through the Integral Sign

Problem: Evaluate $$\int_{0}^{\infty} \dfrac{\sin^3{x}}{x \cdot e^x} dx=\dfrac{A\pi}{B}-\dfrac{\tan^{-1} (C)}{D},$$ My attempt through Differentiation under the Integral Sign: Using $\sin^3x=\frac{3\...
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1answer
65 views

Laplace transform of function

Assume that $f(u)=(\frac{b}{πu^3})^{1/2} e^{2b} e^{-bu} e^{-b/u}$, where $b>0.$ I am trying to calculate the Laplace transform $L\{f(u)\}(s)$ and then the $n_{th}$ derivative of this transform, $L^{...
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1answer
483 views

Evaluating $\int_0^1 \frac{z \log ^2\left(\sqrt{z^2+1}-1\right)}{\sqrt{1-z^2}} \, dz$

What kind of real analysis tools would you employ for this integral? $$\int_0^1 \frac{z \log ^2\left(\sqrt{1+z^2}-1\right)}{\sqrt{1-z^2}} \, dz$$ EDIT: Here is a supplementary question, the cubic ...
2
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2answers
80 views

Convergence of $\int\frac{\arctan x}{x} dx$

I can't find function to bound this integral in the intervals from $1$ to $+\infty$, to prove if it converges. $$ \int _1^{\infty }\frac{\arctan x}{x}dx $$ Any idea? How can I refute this if it is ...
3
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2answers
130 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
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2answers
182 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
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1answer
19 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 ...
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2answers
23 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, ...
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0answers
141 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
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3answers
159 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 ...
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3answers
98 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}} \frac{e^x\sin^2(x)...
5
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3answers
319 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 ...
3
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3answers
67 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 ...
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1answer
30 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
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1answer
65 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 ...
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2answers
60 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 ...
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1answer
40 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 ...
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2answers
79 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 ...
3
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2answers
64 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 ...
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1answer
104 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!
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2answers
47 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?
4
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2answers
128 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 $x=\frac{1-t}{t+1}$ ...
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2answers
64 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) = - \int_0^{\infty}...
11
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3answers
316 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 substitution $x^4 =t$ or $x^4 +x^2+1 =t$ but it makes ...
1
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1answer
57 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 ...
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0answers
59 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 $a\to\...
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0answers
309 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=&\ \color{blue}{-\frac{13\pi^2}{24}\zeta(3)+\frac{47}{2}\zeta(5)-\frac15\ln^52+\frac{\pi^2}9\ln^32-\frac{49\...
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2answers
90 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 ...
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1answer
23 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 ...
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0answers
53 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
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1answer
48 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 ...
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1answer
59 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
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1answer
189 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
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2answers
49 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
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1answer
37 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
77 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
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2answers
39 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
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1answer
30 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
109 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 $\int_{-1}...
0
votes
2answers
66 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
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1answer
56 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 $v = -\frac{1}{a}e^{-ax}$ and ...
4
votes
3answers
97 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 {...