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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0
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0answers
13 views

Finding Fourier coefficient using residue theorem

How i can solve this integral by using residue theorem? $$I = \frac1\pi \int_0^\pi \frac{-\sin(x/2)}{\cos(x/2)} \sin(nx) dx$$
4
votes
1answer
34 views

Existence of improper integral

Prove that $$\int_{0}^{\infty} \frac{(\arctan x)^2}{x^2} dx$$ converges. This is my attempt: The above integral is equal to $$\int_{1}^{\infty} \frac{(\arctan x)^2}{x^2} dx + \int_{0}^{1} ...
1
vote
1answer
42 views

Existence of a function with certain integral properties

Does there exist a non-negative Borel-measurable function $g:\mathbb [1,\infty)\to[0,\infty)$ such that \begin{align*} \int_1^{\infty}g(y)^2\,\mathrm dy<&\,\infty,\\ ...
3
votes
1answer
36 views

Finding a dominating function for this sequence of functions

Problem: Find the limit $$\lim_{n\to\infty} \int_0^n \left( 1 + \frac{x}{n}\right )^{-n} \log(2 + \cos(x/n))dx$$ and justify your reasoning. My Solution: Let $f_n = \left( 1 + \frac{x}{n}\right ...
2
votes
3answers
131 views

A closed form of $\int_0^1\frac{\ln\ln\left(\frac{1}{x}\right)}{x^2-x+1}dx$

This integral has been bugging me since yesterday: $$\int_0^1\frac{\ln\ln\left(\frac{1}{x}\right)}{x^2-x+1}dx$$ I've tried substitution $y=\frac{1}{x}$ and $e^y=\frac{1}{x}$, but those didn't ...
16
votes
5answers
1k views

Simpler way to compute a definite integral without resorting to partial fractions?

I found the method of partial fractions very laborious to solve this definite integral : $$\int_0^\infty \frac{\sqrt[3]{x}}{1 + x^2}\,dx$$ Is there a simpler way to do this ?
25
votes
1answer
1k views

Show that $\int_{0}^{\pi/2}\frac {\log^2\sin x\log^2\cos x}{\cos x\sin x}\mathrm{d}x=\frac14\left( 2\zeta (5)-\zeta(2)\zeta (3)\right)$

Show that : $$ \int_{0}^{\pi/2} {\ln^{2}\left(\vphantom{\large A}\cos\left(x\right)\right) \ln^{2}\left(\vphantom{\large A}\sin\left(x\right)\right) \over ...
6
votes
5answers
179 views

An improper integral : $\int_{0}^\infty {\ln(a^2+x^2)\over{b^2+x^2}}dx$

How to evaluate the following improper integral:$$\int_{0}^\infty {\ln(a^2+x^2)\over{b^2+x^2}}dx,$$ where $a,b>0$. I tried to suppose $$f(a)=\int_0^\infty {\ln(a^2+x^2)\over{b^2+x^2}}dx,$$ based ...
0
votes
2answers
34 views

Convergence of improper integral of $\ln f(x)$

Is there something know about the convergence of $\int_0^1 \ln f(x)dx $ for $f(x)$ continous on $\left(0,1\right)$ and both limits exists, i.e. $\lim_{x\to 0} f(x)$ and $\lim_{x\to 1} f(x)$ ? I ...
2
votes
1answer
704 views

Proving Abel-Dirichlet's test for convergence of improper integrals using Integration by parts

I'm struggling with the following calculus question. Let there be two functions $f,g : [a, \infty) \to \mathbb R$ such that: $g$ is monotonic, differentiable and has a limit at zero $f$ is ...
6
votes
4answers
374 views

A generalized integral need help

I was thinking this integral : $$I(\lambda)=\int_0^{\infty}\frac{\ln ^2x}{x^2+\lambda x+\lambda ^2}\text{d}x$$ What I do is use a Reciprocal subsitution, easy to show that: ...
1
vote
1answer
95 views

Trying to solve $\int{-2\exp{\left(z\cos^2 \theta \frac{\left(a^2 - 1\right)}{2a^2}\right)}}d\theta$

I am trying to solve this integral which has come up as part of some other work, but it is proving to be much harder than I had originally thought. For $0 < |a| \le 1$ being some constant, I am ...
18
votes
3answers
376 views

How to solve $\int_0^\infty J_0(x)\ \text{sinc}(\pi\,x)\ e^{-x}\,\mathrm dx$?

I need some help with solving this integral involving Bessel function: $\hspace{2in}\displaystyle\int_0^\infty$$J_0(x)\ $$\text{sinc}(\pi\,x)\ $$e^{-x}\,\mathrm dx.$
1
vote
1answer
41 views

Improper double integral evaluation by changing the order of integration

I was watching an MIT OCW recitation video about exchanging the order of integration on double integrals. The example was: ...
3
votes
1answer
39 views

Show that $\lim_{s \to \infty}F_s(t) = F(t)$ uniformly for $t \in (0,+\infty)$

Given the following functions: $$ F(t)= \int_0^\infty e^{-tx}\dfrac{\sin{x}}{x}\,dx, \quad t>0$$ $$ F_s(t)= \int_0^s e^{-tx}\dfrac{\sin{x}}{x}\,dx, \quad t \geq 0, s>0$$ Show that $\lim_{s \to ...
2
votes
1answer
86 views

Improper integral $\int_{0}^{\pi} \frac{x}{\sin x} dx$

Find out whether or not the following integral exists $$\int_{0}^{\pi} \frac{x}{\sin x} dx.$$ I'm pretty sure this integral doesn't exist but I can't seem to find a good way to prove this. It ...
11
votes
1answer
320 views

integral $\int_{0}^{\infty}\frac{\cos(\pi x^{2})}{1+2\cosh(\frac{2\pi}{\sqrt{3}}x)}dx=\frac{\sqrt{2}-\sqrt{6}+2}{8}$

Here is a seemingly challenging integral some may try their hand at. $$ \int_{0}^{\infty} {\cos\left(\pi x^{2}\right)\over 1 + 2\cosh\left(\,2\,\pi\,x\,/\,\sqrt{\,3\,}\,\right)}\,{\rm d}x ...
2
votes
2answers
37 views

Fresnel Integral multiplied with cosine term.

$$I=\int_a^b \sin(\alpha-\beta x^2)\cos(x)\, dx.$$ Can anybody tell me, how to solve this integral ? I know that this is related to Fresnel Integral if the $\cos(x)$ term is absent.
23
votes
1answer
465 views

Prove that $\displaystyle\int_0^1{\left\lfloor{1\over x}\right\rfloor}^{-1}\!\!dx={1\over2^2}+{1\over3^2}+{1\over4^2}+\cdots.$

Question. Let $f:[0,1]\to\mathbb R$ given by $$ f(x)=\left\{\,\,\, \begin{array}{ccc} \displaystyle{\left\lfloor{1\over x}\right\rfloor}^{-1}_{\hphantom{|_|}}&\text{if} & 0\lt x\le 1, \\ ...
5
votes
3answers
107 views

How to $\int_{0}^\infty {\sin^3(x)\over x}dx$

How to evaluate : $$\int_{0}^\infty {\sin^3(x)\over x}dx$$ I don't know how to do it. I tried to finish it using integration by parts, but it doesn't work? Can someone tell me how to evaluate the ...
3
votes
5answers
174 views

$\int_0^{+\infty}{ \sin{(ax)} \sin{(bx)}}dx=?$

How can I calculate the integral: $$\int_0^{+\infty}{ \sin{(ax)} \sin{(bx)}}dx$$ ?? I got stuck.. :/ Could you give me some hint?? Do I have to use the following formula?? $\displaystyle{\sin{(A)} ...
5
votes
5answers
207 views

Evaluating the integral $\int_0^\infty \frac{x \sin rx }{a^2+x^2} dx$ using only real analysis

Calculate the integral$$ \int_0^\infty \frac{x \sin rx }{a^2+x^2} dx=\frac{1}{2}\int_{-\infty}^\infty \frac{x \sin rx }{a^2+x^2} dx,\quad a,r \in \mathbb{R}. $$ Edit: I was able to solve the integral ...
1
vote
0answers
152 views

$ \int_0^\infty (1+t^2)^{-s} (1+it)^{s'} 2t \; d t.$

The following integral bothers me since weeks: $$ \int_0^\infty (1+t^2)^{-s} (1+it)^{s'} 2t \; d t.$$ Has any body a suggestion for this integral. $Re\; s >0$ sufficiently large and $s'$ an ...
15
votes
5answers
616 views

Show $ \int_0^\infty\left(1-x\sin\frac 1 x\right)dx = \frac\pi 4 $

$$ \mbox{How to show that }\quad \int_{0}^{\infty}\left[1 - x\sin\left(1 \over x\right)\right]\,{\rm d}x ={\pi \over 4}\quad {\large ?} $$
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votes
1answer
42 views

Value of line integral [closed]

Let $C=\{(x,y)\in \Bbb R^2:~\max\{|x|,|y|\}=1\}.$ The value of the line integral $$\oint_C(xy^2+2y+sin(e^x))dx+(x^2y+cos(e^y))dy$$ is?
2
votes
2answers
37 views

Prove $\int_{-\infty}^{\infty} \frac{du}{(1+u^{2})^{\frac{7-p}{2}}} =\frac{\sqrt{\pi}\Gamma[\frac{1}{2}(6-p)]}{\Gamma[\frac{1}{2}(7-p)]}$

I try to evaluate following integral $\int_{-\infty}^{\infty} \frac{du}{(1+u^{2})^{\frac{7-p}{2}}} =\frac{\sqrt{\pi}\Gamma[\frac{1}{2}(6-p)]}{\Gamma[\frac{1}{2}(7-p)]}$ It seems okay to extend ...
4
votes
7answers
791 views

Integrals from MIT integration bee

$\int\frac{dx}{2+2\sin x+\cos x}$ $\int_0^{\infty}\frac{\ln x}{1+x^2}dx$ $\int\frac{dx}{x(1+x^3)}$ In general what is $\int \frac{dx}{a+b\sin x}$?
2
votes
1answer
38 views

Improper parametric arc length

The first thought I had to solve this problem was using the integral, $$ \int_1^\infty \sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\: \:dt $$ Once you solve for the derivatives ...
7
votes
6answers
399 views

How to calculate $ \int_{0}^{\infty} \frac{ x^2 \log(x) }{1 + x^4} $?

I would like to calculate $$\int_{0}^{\infty} \frac{ x^2 \log(x) }{1 + x^4}$$ by means of the Residue Theorem. This is what I tried so far: We can define a path $\alpha$ that consists of half a ...
10
votes
2answers
373 views

Calculate $\displaystyle \int_0^\infty \frac{\ln x}{1 + x^4} \mathrm{d}x$ using residue calculus

I need to evaluate this integral using calculus of residues: $$\int_0^\infty\frac{\ln(x)}{1+x^4}\mathrm{d}x$$ I know I need to consider $\displaystyle ...
19
votes
8answers
736 views

Integrate: $ \int_0^\infty \frac{\log(x)}{(1+x^2)^2} \, dx $ without using complex analysis methods

Can this integral be solved without using any complex analysis methods: $$ \int_0^\infty \frac{\log(x)}{(1+x^2)^2} \, dx $$ Thanks.
1
vote
0answers
16 views

Approximate improper Bessel integral

In the classic book "Conduction of Heat in Solids" by Carslaw & Jaeger (1959) the heat diffusion equation $$ \frac{\partial T}{\partial t}=\alpha\left(\frac{\partial^2T}{\partial ...
3
votes
2answers
116 views

Find $\lambda$ if $\int^{\infty}_0 \frac{\log(1+x^2)}{(1+x^2)}dx = \lambda \int^1_0 \frac{\log(1+x)}{(1+x^2)}dx$

Problem : If $\displaystyle\int^\infty_0 \frac{\log(1+x^2)}{(1+x^2)}\,dx = \lambda \int^1_0 \frac{\log(1+x)}{(1+x^2)}\,dx$ then find the value of $\lambda$. I am not getting any clue how to proceed ...
9
votes
3answers
147 views

Prove $\int_0^{\infty} \left(\sqrt{1+x^{4}}-x^{2}\right)\ dx=\frac{\Gamma^{2}\left(\frac{1}{4}\right)}{6\sqrt{\pi}}$

I have in trouble for evaluating following integral $$\int_0^{\infty} \left(\sqrt{1+x^{4}}-x^{2}\right)\ dx=\frac{\Gamma^{2}\left(\frac{1}{4}\right)}{6\sqrt{\pi}}$$ It seems really easy, but I ...
0
votes
0answers
41 views

proving improper integral converge

I'm trying to prove the following integral converge: $$ \int_{0}^{\infty}\frac{e^{-\frac{1}{x}}-1}{x^\frac{2}{3}} $$ since 0 and $\infty$ are the problematic points I've done this: $$ ...
2
votes
1answer
26 views

fourier transform of scaled function

let us consider following example one thing which i did not understand is where absolute value of $a$ came from?ok if we have $\int^{\infty}_{-\infty} x(a*t)*e^{-j\omega*t}dt$ then we may have ...
2
votes
2answers
110 views

Closed form for $f(x)=\int_{0}^{+\infty}e^{it^{x}}dt$?

Let $x>1$ and $f(x)=\int_{0}^{+\infty}e^{it^{x}}dt$. Does this integral have a closed form ? Fist point, the integral converges. Indeed let $u=e^{it^{x}}$ and $v=\frac{-i}{x}t^{1-x}$ we have ...
2
votes
1answer
74 views

Laplace transform of : $t^{\gamma-1} F(\alpha,\beta,\delta,\frac{t}{d})$, where $F$ is the Gauss' hypergeometric function

What is the Laplace transform of : $t^{\gamma-1} F(\alpha,\beta,\delta,\frac{t}{d})$, where $\gamma >0 $ and $F$ is the Gauss' hypergeometric function. Note that I have the Laplace transform of : ...
8
votes
3answers
268 views

Proving that $ \displaystyle \gamma = \int_{0}^{1} \!\!\int_{0}^{1} \!\frac{x - 1}{(1 - x y) \log(x y)} \, \mathrm{d}{x} \, \mathrm{d}{y} $.

In 2005, J. Sondow found a surprising formula for the Euler-Mascheroni constant $ \gamma $. The formula is $$ \gamma = \int_{0}^{1} \int_{0}^{1} \frac{x - 1}{(1 - x y) \log(x y)} ~ \mathrm{d}{x} ~ ...
3
votes
0answers
29 views

Parameter-dependent integral: Is the following statement true?

Is the following statement true? If so, could anyone provide a reference? Suppose $f(x, \alpha)$ is continuous on $(a, b) \times \{\alpha_0\}$. If there exists $g(x)$ which is continuous on $(a, b)$, ...
6
votes
4answers
467 views

Integrate: $\int_0^\infty \frac{\log (1+x)}{1+x^2}dx$

Can this integral be solved with contour integral or by some application of Residue theorem? $$\int_0^\infty \frac{\log (1+x)}{1+x^2}dx = \frac{\pi}{4}\log 2 + \text{Catlan constant}$$ It has two ...
5
votes
5answers
510 views

$\int_0^\infty\frac{\log x dx}{x^2-1}$ with a hint.

I have to calculate $$\int_0^\infty\frac{\log x dx}{x^2-1},$$ and the hint is to integrate $\frac{\log z}{z^2-1}$ over the boundary of the domain $$\{z\,:\,r<|z|<R,\,\Re (z)>0,\,\Im ...
13
votes
2answers
427 views

Interesting Integral $\int_{-\infty}^{\infty}\frac{e^{i nx}}{\Gamma(\alpha+x) \Gamma(\beta -x)}dx$

I am asking this question out of curiosity. $$\int_{-\infty}^{\infty}\frac{e^{i nx}}{\Gamma(\alpha+x) \Gamma(\beta -x)}dx = \frac{ \left(2\cos \frac{n}{2} \right)^{\alpha ...
1
vote
3answers
29 views

Growth restriction for nonnegative, continuous functions whose integrals on $\mathbb{R}$ are bounded

When we have a nonnegative, continuous function $f(x)$ whose integral over all real numbers $\mathbb{R}$ is bounded, like: $$\int_{-\infty}^{\infty}f(x)dx = A< \infty $$ with $A \in \mathbb{R}$ ...
1
vote
0answers
52 views

Question on integral

I need a confirmation and answer about the following problem: If we have $ g(x)=\ln x+{ x }^{ -1/2 }{ 1 }_{ x\le 1 } $ I'm trying to determine $ \int _{ 0 }^{ +\infty }{ \ln x+{ x }^{ -1/2 }{ 1 }_{ ...
1
vote
1answer
252 views
4
votes
3answers
80 views

Evaluate Gauss-like Integral

Evaluate Integral $$\int_0^\infty e^{-ay^{2}-\frac{b}{y^2}}dy $$ Where a and b are real and positive. This integral is eerily similar to the Gaussian integral $$\int_0^\infty e^{-\alpha x^2}dx = ...
0
votes
1answer
27 views

How to identify continuity or discontinuity of an [Definite] integral?

How can I figure out whether an improper integral converges based on the discontinuities in the integrand? For instance, these two both have discontinuities within the intervals of integration, and ...
0
votes
1answer
63 views

a question about integral proof: $\lim_{n\to \infty} \int_{0}^\infty {n\cdot {\ln(1+{f(x)\over n}}})dx=\int_{0}^\infty f(x)dx$

A non-negative function ${\rm f}\left(x\right)$ is continuous in $(0,\infty)$ and $\displaystyle{\int_{0}^{\infty}{\rm f}\left(x\right)\,{\rm d}x}$ is convergent. Then, we need to prove $$\lim_{n\to ...
4
votes
1answer
75 views

a complicated question about double improper integral

how to evaluate $$\iint_{y\ge x^2+1}{dx\,dy\over{x^4+y^2}}$$ My solution: the initial intergral $$ =2\int_0^\infty \left(\int_{x^2+1}^\infty {dy\over {x^4+y^2}}\right)\,dx = \int_0^\infty ...