Questions tagged [improper-integrals]

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 $\infty$ or $-\infty$, or as both endpoints approach limits.

Filter by
Sorted by
Tagged with
-4
votes
2answers
71 views

Prove $\lim_{\epsilon\to 0^{+}}\int_{\epsilon}^{\infty}f(x){\rm d}x= \int_{0}^{\infty}f(x){\rm d}x $ where $f$ is integrable in $[0,\infty)$

If $f$ is integrable in $[0,\infty)$ then $$\lim_{\epsilon\to 0^{+}}\int_{\epsilon}^{\infty}f(x){\rm d}x= \int_{0}^{\infty}f(x){\rm d}x$$ Attempt $$|\int_{\epsilon}^{\infty}f(x){\rm d}x- \int_{0}^{\...
2
votes
1answer
56 views

Limits of an Improper Integral: $\lim\limits_{n\to\infty} \int_{-\infty}^{+\infty}f(x)\sin(nx)dx$

Is $$ \lim\limits_{n\to\infty} \int_{-\infty}^{+\infty}f(x)\sin(nx)dx=0 $$ when $\int_{-\infty}^{+\infty}f(x)dx$ is convergent but not absolutely convergent?
1
vote
0answers
44 views

Is There an Abel-plana-type Exact Formula for the Imaginary Part of the Oscillatory Integral over Exp(iπx)x^(1/x) Between 1 and Infinity?

This is a question that is an important part of my life-long hobby of researching the MRB constant. CMRB I have posted Mathematica code with all my formulas below. Richard J. Mathar did in-depth ...
0
votes
1answer
27 views

Improper Integral Calculation with Lots of Constants

I have an improper integral: $T=4\sqrt{m}\int_0^\sqrt{2E/k}\frac{1}{\sqrt{2E-kx^2}}dx$ I said it was improper because if you plug in the upper bound, I believe the denominator = 0. I am supposed to ...
0
votes
1answer
12 views

Improper Integral Convergence Properties [closed]

Let $f,g:(0,\infty) \to \mathbb{R}$ be continuous function with $f,g>0$ on $(0,\infty)$. a) If $\int_1^{\infty}f(x) dx$ converges, then $\int_0^{1}\frac{1}{f(x)} dx$ converges. b) If $\int_1^{\...
0
votes
1answer
21 views

Convergence exercise related to improper integral

I'm doing the following exercise in mathematical analysis: (about improper integral) Let $f\in C^1(0,1]$ be a continuously differentiable function on the half interval $(0,1]$. Suppose $f$ is ...
6
votes
2answers
224 views

Derive $\int_0^1 \frac{\ln(\sqrt2-1)-(\sqrt2-x)\ln x}{(\sqrt2-x)^2-1}\,dx=\frac{\pi^2}6+\frac14\ln^2(\sqrt2-1) $

I obtained the integral $$\int_0^1 \frac{\ln(\sqrt2-1)-\ln(x)(\sqrt2-x)}{(\sqrt2-x)^2-1}\,dx=\frac{\pi^2}6+\frac14\ln^2(\sqrt2-1) $$ as a by-product while carrying out some complex analysis on an ...
0
votes
0answers
38 views

Finding $p,q$ as those integrals converge? [closed]

$$\begin{align*} {I}_1&=\int_0^\infty \frac{\ln x}{1+x^p}{\rm d}x\\[5pt] {I}_2&=\int_0^\infty \frac{x^q\ln^2 x }{1+x^p}{\rm d}x\\[5pt] \end{align*}$$ Does anyone know how can I find $p,q$ as ...
2
votes
5answers
80 views

Solve for $p$ in $\int_{0}^{1} \frac{1}{x^p}\,dx = \frac{4}{3}$

I did a question $\int_{0}^{1}\frac{1}{x^{\frac{1}{2}}}\,dx$, and evaluating this is divergent integral yes? Then as a general form $\int_{0}^{1} \frac{1}{x^p}\,dx$, $p \in \mathbb{R}$, what values of ...
0
votes
0answers
38 views

Indefinite multiple integral

Let: $\displaystyle F(x,y) = \int \int \frac{1}{xy} \, dx \, dy$ be a bidimensional indefinite integral. I would like to understand the correct way to handle it. My first attempt would be to simply ...
1
vote
0answers
37 views

Convergence of improper integral $\int_0^{\infty} 1/(1+x^4 \sin^2(x)) \,dx$ [duplicate]

I want to determine the convergence of the following improper integral:$$\int_0^{\infty} \frac{1}{1+x^4\sin^2x}\,dx.$$ I have tried comparison tests for the same. $$\int_0^{\infty} \frac{1}{1+x^4}\,...
-3
votes
0answers
94 views

How do we find the integrals of the kind $\int_0^\infty (1-\frac1{x^2})dx$ or $\int_0^\infty (x-\frac1{x^3})dx$? [closed]

Of course, all these are zero, but how do we know this? What are the used techniques? $\int_0^\infty (1-\frac1{x^2})dx=0$ $\int_0^\infty (x-\frac2{x^3})dx=0$ $\int_0^\infty (x^2-\frac6{x^4})dx=0$ etc. ...
0
votes
3answers
98 views

Stuck with integral involving Polylogarithms $\int_{0}^{\infty}\frac{2t}{e^{t\pi}+1} \,dt$

For a research work I ended up needing to give a proof of Zeta's trivial zeros, in order to do so I tried using the Abel-Plana formula. $\zeta(s)=\frac{2^{s-1}}{s-1}-2^{s}\int_{0}^{\infty} \frac{\sin(...
0
votes
1answer
66 views

How to approximately calculate the integration : $ I = \int_{0}^{\infty}(\frac{1}{x^2+a^2})(\frac{1}{(x-1)^2+b^2}) dx $ [closed]

Integration : $ I = \int_{0}^{\infty}(\frac{1}{x^2+a^2})(\frac{1}{(x-1)^2+b^2}) dx $ when : (i) $a\ll 1$, $b\sim 1$; (ii) $a=b\gg 1$ I want some hints for the transformation of the parameters or ...
0
votes
1answer
31 views

The Lebesgue Monotone Convergence theorem on function domains

I am trying to prove an analogous statement of the Lebesgue Monotone Convergence theorem on function domains. Let $A \subset \mathbb{R}^n$ be a bounded set that has volume. Let $f : A \mapsto \mathbb{...
0
votes
2answers
90 views

Need a good method for computing $\int _{\infty }^a\:\frac{1}{\left(x^2-1\right)^p}dx$

I need a good method for computing $$\int _{\infty }^a\:\frac{1}{\left(x^2-1\right)^p}dx$$ Where $p>0$ and $a>1$ are real. Maybe you have a good numerical algorithm for such a computation? That ...
0
votes
0answers
37 views

Prove or disprove Catalan improper integral

Prove or disprove that the improper integral $$ I = \int_{0}^{\infty}\frac{e^{-x}\sin x}{\arctan x}dx = G $$ $G =$ Catalan constant. By Wolfram Alpha, $G - I =0.0000140883$.
0
votes
0answers
47 views

How can we get the domain of $F(x) = \int f(x) dx$?

Let $$F(x) = \int f(x) dx.$$ My question is how can we get the domain of F(x) generally. I usually think in f(x), and I imagine where it makes sense to take the area under the curve, but I'm afraid ...
1
vote
1answer
29 views

How to show that $\int_{-\infty}^{\infty}|\frac{\sin{nx}\sin{x}}{x^2}|dx \to \infty$ as $n \to \infty$? [duplicate]

We now have a sequence of functions \[ f_n = \frac{\sin{nx}\sin{x}}{x^2}, \] and we need to show that $\lVert f_n \rVert_1 \to \infty$ as $n \to \infty$, which means \[ \lim_{n \to \infty}\int_{-\...
2
votes
0answers
41 views

Exercise 5.S. the elements of integration and lebesgue measure

I'm having problems with this exercise. I've tried to apply the DOMINATED CONVERGENCE THEOREM but I couldn't. Could someone gives me any hint? Suppose the function $x\rightarrow f(x,t)$ is $X$-...
1
vote
1answer
35 views

How can I find the domain of $ \int_{0}^{x} \frac{\log(1-t^{2})+t^{2}}{t}dt$?

Let $$ h(x) = \int_{0}^{x} \frac{\log(1-t^{2})+t^{2}}{t}dt \ .$$ How can I find the domain of this function? I tried to integrate it, but it seems impossible; I also thought that it could be the ...
0
votes
1answer
42 views

Computing a singular integral

I would like to derive the exact value of the following integral $$ I_s= \int_1^\infty r| r^{-2s}- (r^2-1)^{-s}|d r\qquad \text{with}\qquad 0<s<1.$$ The existence of $I_s$ is warranted since ...
8
votes
0answers
76 views

Evaluating $\int_{1}^{\infty} \frac{1}{\lfloor{x}\rfloor!}dx$

I thought the improper integral $\int_{1}^{\infty} \frac{1}{\lfloor{x}\rfloor!}dx$ converge, while the textbook says it's not. Of course, $\lfloor{x}\rfloor$ is the greatest integer function. Here is ...
0
votes
0answers
18 views

Integral of multinormal distribution of 3 variables

I am interested in the integral $P(f_i \leq f \leq f_j)$ where each function follows a normal distribution with distinct parameters. With a bit of algebra, I managed to show that this integral is ...
4
votes
1answer
63 views

Integrating $\log(-ix)\exp(-ix)/x^2$

I would like to compute a few integrals like $$\int_{-\infty}^\infty\frac{\log(-ix)\exp(-ix)}{x^2}\,dx$$ To be clear, here the path of integration is really $z = \epsilon i + x$, so that it avoids the ...
5
votes
3answers
240 views

Evaluate an absolute monster integral $\int\limits_{0}^{1} \frac{\log(1-x+x^2)}{\sqrt{x}(1+x)}\mathrm{d}x.$

I want to figure out a way to evaluate $$\int\limits_{0}^{1} \frac{\log(1-x+x^2)}{\sqrt{x}(1+x)}\mathrm{d}x.$$ I tried to substitute $x = u^2$ and cancel the square root in the denominator, getting $$...
2
votes
1answer
90 views

$\int_0^1\frac{\ln^2x\ln(1+x)}{1-x}\,dx$

Someone please help me to solve this integral $$\int_0^1\frac{\ln^2x\ln(1+x)}{1-x}\,dx$$ I have tried to use the formula $$ab^2=\frac{(a+b)^3+(a-b)^3+2a^2}{6}$$ to reduce the original integral to ...
0
votes
1answer
47 views

Comparison Test for $-\ln\left(1-\dfrac{\pi^2}{2x^2}\right)$

I'm trying to use the comparison test to show that $\displaystyle \int_3^\infty -\ln(1-\frac{\pi^2}{2x^2})$ converges, but I'm having trouble finding a function that is greater than it and still ...
0
votes
3answers
86 views

If $\int_{0}^{\infty} f(x)dx$ converges, $f(x)$ is differentiable on $[0,\infty)$, does $\int_{0}^{\infty} f'(x)dx$ also converge?

If $\int_{0}^{\infty} f(x)dx$ converges, and $f(x)$ is differentiable on $[0,\infty)$, does $\int_{0}^{\infty} f'(x)dx$ also converge?
1
vote
1answer
65 views

Integral of two error functions times a Gaussian

I am trying to find some expression for this integral, ideally as a function of the constants. I can't put the full integral in the title as it's too long. I doubt a closed form exists, but I'm even ...
0
votes
2answers
34 views

Convergence of the logarithmic integral $\int_{0.5}^1\frac 1{\ln(x)} \,dx$

I computed the integral using Wolfram Alpha, and it gave me values in terms of the logarithmic integral, but I am unsure on how to prove convergence or divergence using analysis. So I want to prove ...
2
votes
2answers
97 views

If $\int_a^\infty f(x)dx$ exists (converges), then $\int_a^\infty \frac{f(x)}{x}dx$ exists

This is a true or false statement: If $\displaystyle \int_a^\infty f(x)dx$ exists (converges), then $\displaystyle\int_a^\infty \frac{f(x)}{x}dx$ converges for $a>0$ (and $a\in\mathbb{R^+}$). In ...
0
votes
1answer
49 views

Improper Integral via Residue $\int_0^\infty \frac{x^{a-1}\cos(\log_e x)}{e^x+1}dx , 0<a<1$

Evaluate by Residue theory $$\int_0^\infty \frac{x^{a-1}\cos(\log_e x)}{e^x+1}dx , \quad 0<a<1$$ My try - $$I= \int_0^\infty \frac{x^{a-1}\cos(\log_e x)}{e^x+1}dx ,\quad 0<a<1$$ $$I= \...
0
votes
1answer
50 views

Integral of product of Gamma, Erfc and Exponential

I can confirm that: $$\int_{-\infty }^{\infty } \frac{e^{-\frac{K^2}{2}} 2^{-n-\frac{1}{2}} n \Gamma \left(\frac{1}{2},\frac{K^2}{2}\right) \text{erfc}\left(\frac{-K}{\sqrt{2}}\right)^{n-1}}{\pi } \, ...
-2
votes
1answer
61 views

Divergence of $I= \lim_{n \to \infty} \int_{\ln (n)}^{\ln(n+1)}(e^{\frac{3x}{4}}-e^{\frac{x}{4}})\sin(x) \mathrm dx$ [closed]

$$I= \lim_{n \to \infty} \int_{\ln (n)}^{\ln(n+1)}(e^{\frac{3x}{4}}-e^{\frac{x}{4}})\sin(x)\mathrm dx$$ Prove that either $I$ is divergent or $I\neq 0$ My try $\displaystyle \int e^{ax}\sin (bx) \...
-2
votes
3answers
41 views

Approach to test convergence of $\int_1^\infty \frac{\sqrt{x+1}}{x^2} dx$ [closed]

The following improper integral converges: $\int_1^\infty \frac{\sqrt{x+1}}{x^2} dx$ This integral CAN be evaluated, however is there a simpler way to test for convergence/divergence without ...
0
votes
1answer
42 views

Studying the nature of an improper integral

How can I determine the nature of this integral and also, how can I calculate it? I've been trying to do it for like an hour, but I can't figure it out. $$ \int_{1}^{\infty} \frac{\cos x \ln x}{x} dx $...
1
vote
1answer
141 views

Prove $\int_{2}^{\infty}\ln\sec\frac{\pi}{x}\,dx$ is convergent

I have proved that $\int_{2}^{3}\ln\sec\frac{\pi}{x}\,dx$ is convergent by first proving that $$\sec x\leq\frac{1}{1-\frac{2}{\pi}x}$$ and therefore proving that if $\int_{2}^{3}\frac{1}{1-\frac{2}{x}...
9
votes
7answers
346 views

Prove $\int_0^\infty\frac{\ln x}{x^3-1}\mathrm{d}x=\frac{4\pi^2}{27}$

Proof of the integral $$\int_0^\infty\frac{\ln x}{x^3-1}\mathrm{d}x=\frac{4\pi^2}{27}$$ I try to substitute $u = \ln x$. Then $x = e^u,\>\mathrm{d}x = e^u\mathrm{d}u$ and the limits $(0,\infty)\to (...
0
votes
0answers
81 views

Evaluation of $\zeta$ for integer values

I have recently found that $\zeta(2)$ can be found by integrating $\frac{\log(1+x)}{x}$ from $-1$ to $1$. Since $$ \int_{-1}^{1}\frac{\log(1+x)}{x}dx=\int_{-1}^{1}(1-\frac{x}{2}+\frac{x^2}{3}-\frac{x^...
2
votes
2answers
230 views

Very difficult double integral

How does one solve this integral ? $$\iint_{-\infty}^\infty \frac{1}{4\pi^2}e^{iu(x-x'-\varepsilon\Delta_{})}e^{iv(z-z'-\varepsilon\Delta_2)} e^{-\varepsilon bv^2}e^{-\varepsilon au^2}e^{-\varepsilon ...
3
votes
2answers
77 views

Show that $\lim_{n \to \infty} (\int_{0}^{\infty}f_n(x)dx) \not= \int_0^\infty f(x)dx$

There is a sequence of functions, $f_{1},f_{2},f_{3},\ldots$, whose domain is $[0, \infty)$, such that $$f_{n}(x)=\begin{cases} \frac{1}{n} & \text{if } x \in [0, n) \\ \frac{-1}{n}x+...
-1
votes
0answers
23 views

relation between absolute convergence of a series and improper integral

I'm trying to answer to some questions in preparation to an exam and I found this one that is quite strange. It asks me to "show the relation between absolute convergence of a series and improper ...
7
votes
2answers
104 views

Show $\int_0^\infty \frac{\tan^{-1}x^2}{1+x^2} dx= \int_0^\infty \frac{\tan^{-1}x^{1/2} }{1+x^2}dx$

I accidentally found out that the two integrals below $$I_1=\int_0^\infty \frac{\tan^{-1}x^2}{1+x^2} dx,\>\>\>\>\>\>\>I_2=\int_0^\infty \frac{\tan^{-1}x^{1/2} }{1+x^2}dx$$ are ...
5
votes
0answers
46 views

Is it wrong to use limits to calculate an integral of a Piecewise Function?

In my test I used limits when I calculated the integral of a piecewise function. In my head the reasoning was that it was an improper integral in one piece of the function because it never reached ...
1
vote
0answers
81 views

Finding the Antiderivative of $x^{-2}(1+x^{-1})^x$

I'm currently working on solving the following limit/integral: $$\lim_{t\to\infty}\int_0^t{\frac 1{x^2}\left(1+\frac 1x\right)^{-x}dx}$$ Here's what I've established so far: The integral converges. -...
0
votes
1answer
39 views

Is there any way to evaluate/ estimate this integral apart from wolfram? [duplicate]

I am looking for a way to evaluate (if not, then estimate) the following $$\int_0^x \dfrac {1}{\sqrt{\cos t - \cos x}}dt, \ 0 < x \leq \pi$$ Wolfram tells me that it blows up at $x = \pi$ and for ...
1
vote
2answers
57 views

Determine with proof if the following integral converges

Determine if the following integral converge. $$\int^\infty_2 \ln\left(\sec\left(\pi\over x\right)\right)\ dx$$ I have a function to compare the above over the interval [2, 3], but am struggling to ...
0
votes
0answers
24 views

why the value of integral conflicts with the graph?

considering the graph of 1/x and 1/x^2, both of the function’s integration from x=1 to x=infinity looks like a finite number but 1/x is not finite. I know mathematicly this is correct but why one is ...
1
vote
1answer
41 views

Improper integrals involving logarithm in denominator

Discuss/comment on the convergence/divergence of the following improper integrals: (A) $$\int_0^{+\infty}\dfrac{e^{-x}}{x^{\alpha}(|\log x|)^{\beta}},\text{ with }\alpha,\beta>0$$ (B) $$\int_0^{+\...

1
2 3 4 5
129