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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58
votes
14answers
9k views

Proving $\int_{0}^{\infty} \mathrm{e}^{-x^2} dx = \dfrac{\sqrt \pi}{2}$

How to prove $$\int_{0}^{\infty} \mathrm{e}^{-x^2}\, dx = \frac{\sqrt \pi}{2}$$
75
votes
17answers
16k 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 anyone ...
42
votes
5answers
2k views

Calculating the integral $\int_{0}^{\infty} \frac{\cos x}{1+x^2}\mathrm{d}x$ without using complex analysis

Suppose that we do not know anything about the complex analysis (numbers). In this case, how to calculate the following integral in closed form? $$\int_0^\infty\frac{\cos\;x}{1+x^2}\mathrm{d}x$$
2
votes
1answer
531 views

A few improper integral

$$\displaystyle \begin{align*} & \int_{0}^{+\infty }{\frac{\text{d}x}{1+{{x}^{n}}}} \\ & \int_{-\infty }^{+\infty }{\frac{{{x}^{2m}}}{1+{{x}^{2n}}}\text{d}x} \\ & \int_{0}^{+\infty ...
19
votes
6answers
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 ?
39
votes
3answers
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}^{\Large\frac\pi2} {\ln^{2}\left(\vphantom{\large A}\cos\left(x\right)\right) \ln^{2}\left(\vphantom{\large A}\sin\left(x\right)\right) \over ...
26
votes
6answers
7k views

Does $ \int_0^{\infty}\frac{\sin x}{x}dx $ have an improper Riemann integral or a Lebesgue integral?

In this wikipedia article for improper integral, $$ \int_0^{\infty}\frac{\sin x}{x}dx $$ is given as an example for the integrals that have an improper Riemann integral but do not have a (proper) ...
11
votes
5answers
807 views

$\int_0^{\infty}\frac{\ln x}{x^2+a^2}\mathrm{d}x$ Evaluate Integral

Anyone remember the method for this? I think this should been done on the site $$\int_0^{\infty}\frac{\ln x}{x^2+a^2}\mathrm{d}x$$
18
votes
4answers
792 views

Prove $\int_0^{\infty}\! \frac{\mathbb{d}x}{1+x^n}=\frac{\pi}{n \sin\frac{\pi}{n}}$ using real analysis techniques only

I have found a proof using complex analysis techniques (contour integral, residue theorem, etc.) that shows $$\int_0^{\infty}\! \frac{\mathbb{d}x}{1+x^n}=\frac{\pi}{n \sin\frac{\pi}{n}}$$ for $n\in ...
14
votes
4answers
4k views

Prove: $\int_{0}^{\infty} \sin (x^2) dx$ converges.

$\sin(x^2)$ is an example for a function which its limit when $x \to \infty$ is not $0$, and still its integral from $0$ to $\infty$ is finite. I'd like your help with understanding why and a ...
13
votes
7answers
694 views

Evaluating $\int_0^\infty \frac{dx}{1+x^4}$. [duplicate]

Can anyone give me a hint to evaluate this integral? $$\int_0^\infty \frac{dx}{1+x^4}$$ I know it will involve the gamma function, but how?
4
votes
2answers
434 views

Principal value of the singular integral $\int_0^\pi \frac{\cos nt}{\cos t - \cos A} dt$

For a constant $0<A<\pi$, and natural $n$ I want to find the principal value of the integral: $$\int_0^\pi \frac{\cos nt}{\cos t - \cos A} dt$$ First of all, I'm not certain what function in ...
28
votes
3answers
890 views

$\int_{0}^{\infty} \frac{\cos x - e^{-x^2}}{x} \ dx$ Evaluate Integral

Evaluate $$\int_{0}^{\infty} \frac{\cos x - e^{-x^2}}{x} \ dx$$
7
votes
5answers
462 views

Definite integral, quotient of logarithm and polynomial

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: ...
6
votes
2answers
566 views

Laplace transform:$\int_0^\infty \frac{\sin^4 x}{x^3} \, dx $

I have a trouble with a integral: Using this Laplace trasform equation: $$\begin{align} \int_0^\infty F(u)g(u) \, du & = \int_0^\infty f(u)G(u) \, du \\[6pt] L[f(t)] & = F(s) \\[6pt] ...
57
votes
6answers
2k views

How to find ${\large\int}_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx$

Please help me to find a closed form for this integral: $$I=\int_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx\tag1$$ I suspect it might exist because there are similar integrals having closed forms: ...
27
votes
3answers
726 views

What is $\int_0^1\frac{x^7-1}{\log(x)}\mathrm dx$?

/A problem from the 2012 MIT Integration Bee is $$ \int_0^1\frac{x^7-1}{\log(x)}\mathrm dx $$ The answer is $\log(8)$. Worlfram alpha gives an indefinite form in terms of the logarithmic integral ...
18
votes
5answers
1k views

Evaluating $\int\limits_0^\infty \! \frac{x^{1/n}}{1+x^2} \ \mathrm{d}x$

I've been trying to evaluate the following integral from the 2011 Harvard PhD Qualifying Exam. For all $n\in\mathbb{N}^+$ in general: $$\int\limits_0^\infty \! \frac{x^{1/n}}{1+x^2} \ \mathrm{d}x$$ ...
15
votes
4answers
597 views

Looking for closed-forms of $\int_0^{\pi/4}\ln^2(\sin x)\,dx$ and $\int_0^{\pi/4}\ln^2(\cos x)\,dx$

A few days ago, I posted the following problems Prove that \begin{equation} \int_0^{\pi/2}\ln^2(\cos x)\,dx=\frac{\pi}{2}\ln^2 2+\frac{\pi^3}{24}\\[20pt] -\int_0^{\pi/2}\ln^3(\cos ...
14
votes
2answers
635 views

How to evaluate these integrals by hand

How can I evaluate$$\int_{-\infty}^\infty \frac{\cos x}{\cosh x}\,\mathrm dx\text{ and }\int_0^\infty\frac{\sin x}{e^x-1}\,\mathrm dx.$$ Thanks in advance.
15
votes
7answers
561 views

Evaluate $\int_0^\infty\frac{\ln x}{1+x^2}dx$

Evaluate $$\int_0^\infty\frac{\ln x}{1+x^2}\ dx$$ I don't know where to start with this so either the full evaluation or any hints or pushes in the right direction would be appreciated. Thanks.
1
vote
2answers
3k views

Calculate the Fourier transform of ${\rm b}\left(x\right) = 1/\left(x^{2} +a^{2}\right)$

I need help to calculate the Fourier transform of this funcion $${\rm b}\left(x\right)=\frac{1}{x^{2} + a^{2}}\,,\qquad a > 0$$ Thanks.
33
votes
8answers
2k views

Why do we restrict the definition of Lebesgue Integrability?

The function $f(x) = \sin(x)/x$ is Riemann Integrable from $0$ to $\infty$, but it is not Lebesgue Integrable on that same interval. (Note, it is not absolutely Riemann Integrable.) Why is it we ...
25
votes
3answers
548 views

Integral of Combination Log and Inverse Trig Function

Does the following integral have a closed-form ?: \begin{equation} \int_{0}^{1}{\ln\left(\,x\,\right) \over 1 + x}\,\arccos\left(\,x\,\right) \,{\rm d}x \end{equation} This integral has been ...
11
votes
1answer
1k views

Calculating the Fourier transform of $\frac{\sinh(kx)}{\sinh(x)}$

I'm trying to compute $$\int_{-\infty}^\infty \frac{\sinh(kx)}{\sinh(x)}e^{-i\omega x} \ dx$$ i.e. the Fourier transform of $x\mapsto \frac{\sinh(kx)}{\sinh(x)}$, where $0<k<1$ is fixed. But ...
2
votes
2answers
412 views

Calculating: $\lim_{n\to \infty}\int_0^\sqrt{n} {(1-\frac{x^2}{n})^n}dx$ [duplicate]

Possible Duplicate: Prove: $\lim\limits_{n \to \infty} \int_{0}^{\sqrt n}(1-\frac{x^2}{n})^ndx=\int_{0}^{\infty} e^{-x^2}dx$ I need some help calculating the above limit. What i have ...
29
votes
2answers
1k views

Integral $\int_0^1\frac{1-x^2+\left(1+x^2\right)\ln x}{\left(x+x^2\right)\ln^3x}dx$

I'm struggling with this integral $$I=\int_0^1\frac{1-x^2+\left(1+x^2\right)\ln x}{\left(x+x^2\right)\ln^3x}dx.\tag1$$ Mathematica could not evaluate it in a closed form. Its numeric value is ...
20
votes
5answers
3k views

How to find the integral $\int_{0}^{\infty}\exp(- (ax+b/a))\,dx$?

How do I find $$\large\int_{0}^{\infty}e^{-\left(ax+\frac{b}{x}\right)}dx$$ where $a$ and $b$ are positive numbers? This is not a homework question. I will be quite happy if somebody can come up ...
14
votes
2answers
311 views

$\int_0^\infty(\log x)^2(\mathrm{sech}\,x)^2\mathrm dx$

Is there any closed-form representation for the following integral? $$\int_0^\infty(\log x)^2(\mathrm{sech}\,x)^2\mathrm dx,$$ where $\mathrm{sech}\,x$ is the hyperbolic secant, ...
28
votes
4answers
855 views

Evaluating $\int_{0}^{1}\frac{1-x}{1+x}\frac{\mathrm dx}{\ln x}$

Some time ago I came across to the following integral: $$I=\int_{0}^{1}\frac{1-x}{1+x}\frac{\mathrm dx}{\ln x}$$ What are the hints on how to compute this integral?
7
votes
4answers
231 views

How to evaluate $\int_0^\infty \frac{1}{x^n+1} dx$

Noticed that the integral $$\int_0^\infty \frac{1}{x^n+1} dx$$ is often approached with partial fraction decomposition, but this gets increasingly ugly as $n$ gets bigger. Is there a neat trick to do ...
8
votes
3answers
240 views

Evaluate $\int^1_0 \log^2(1-x) \log^2(x) \, dx$

I have no idea where to even start. WolframAlpha cant compute it either. $$\int^1_0 \log^2(1-x) \log^2(x) \, dx$$ I think it can be done with series, but I am not sure, can someone help a little so ...
6
votes
5answers
683 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 ...
3
votes
1answer
1k 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 ...
0
votes
1answer
44 views

Generalized Riemann Integral: Improper Version

Reference For a bounded nonexample of integrability see: Riemann Integral: Bounded Nonexample For a convergence theorem on integral see: Riemann Integral: Uniform Convergence For a comparison of ...
30
votes
8answers
1k views

Evaluate $\int_0^1\left(\frac{1}{\ln x} + \frac{1}{1-x}\right)^2 \mathrm dx$

Evaluate $$\int_0^1\left(\frac{1}{\ln x} + \frac{1}{1-x}\right)^2 \mathrm dx$$
27
votes
3answers
622 views

Prove $\displaystyle \int_{0}^{\pi/2} \ln \left(x^{2} + (\ln\cos x)^2 \right) \, dx=\pi\ln\ln2 $

How to prove $$ \int_{0}^{\pi/2}\ln\left(\,x^{2} + \ln^{2}\left(\,\cos\left(\,x\,\right)\,\right) \,\right)\,{\rm d}x\ =\ \pi\ln\left(\,\ln\left(\, 2\,\right)\,\right) $$ I don't know how to ...
20
votes
8answers
998 views

A proof of $\int_{0}^{1}\left( \frac{\ln t}{1-t}\right)^2\,\mathrm{d}t=\frac{\pi^2}{3}$

What is the proof of the following: $$\int_{0}^{1} \left(\frac{\ln t}{1-t}\right)^2 \,\mathrm{d}t=\frac{\pi^2}{3} \>?$$
22
votes
4answers
618 views

How to prove $\int_{-\infty}^{+\infty} f(x)dx = \int_{-\infty}^{+\infty} f\left(x - \frac{1}{x}\right)dx?$

If $f(x)$ is a continuous function on $(-\infty, +\infty)$ and $\int_{-\infty}^{+\infty} f(x)dx$ exists. How can I prove that $$\int_{-\infty}^{+\infty} f(x)dx = \int_{-\infty}^{+\infty} f\left( x - ...
13
votes
5answers
592 views

Evaluate the integral $\int_{0}^{\infty} \frac{1}{(1+x^2)\cosh{(ax)}}dx$

The problem is : Evaluate the integral $$\int_{0}^{\infty} \frac{1}{(1+x^2)\cosh{(ax)}}dx$$ I have tried expand $\frac{1}{\cosh{ax}}$ and give the result in the following way: First, note ...
15
votes
8answers
1k views

$\int_{0}^{\infty} \frac{e^{-x} \sin(x)}{x} dx$ Evaluate Integral

Compute the following integral: $$\int_{0}^{\infty} \frac{e^{-x} \sin(x)}{x} dx$$ Any hint, suggestion is welcome. Thanks.
13
votes
2answers
438 views

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

This integral has been bugging me since yesterday: $$\int_0^1\frac{\ln\ln\left({1}/{x}\right)}{x^2-x+1}\mathrm dx$$ I've tried substitution $y={1}/{x}$ and $e^y={1}/{x}$, but those didn't help ...
6
votes
4answers
368 views

How to compute $\int_{-\infty}^\infty\exp\left(-\frac{(x^2-13x-1)^2}{611x^2}\right)\ dx$

$$\int_{-\infty}^\infty\exp\left(-\frac{(x^2-13x-1)^2}{611x^2}\right)\ dx$$ WolframAlpha gives a numerical answer of $43.8122$, which appears to be $\sqrt{611\pi}$. And playing with that, it seems ...
6
votes
1answer
5k views

Improper integral sin(x)/x converges absolutely, conditionaly or diverges?

$$\int_1^{\infty}\frac{\sin x}{x}dx$$ $$u=\frac{1}{x}$$ $$du=-\frac{1}{x^2}dx$$ $$dv=\sin xdx$$ $$v=-\cos x$$ $$\int_1^{\infty}\frac{\sin x}{x}dx=\frac{1}{x}(-\cos x)-\int_1^{\infty}\frac{\cos ...
2
votes
0answers
118 views

integral involving upper incomplete gamma function

I trying to evaluate the following integral $$\int_0^\infty \dfrac { x^{m-1} \Gamma(A,\mathcal B x^q)} {\left[1+(\eta x)^n\right]^p} \,\mathrm dx$$ where the integration is w.r.t. $x$, and the ...
11
votes
6answers
241 views
11
votes
5answers
537 views

How to evaluate $\int_{0}^{+\infty}\exp(-ax^2-\frac b{x^2})\,dx$ for $a,b>0$

How can I evaluate $$I=\int_{0}^{+\infty}\!e^{\left(-ax^2-\frac b{x^2}\right)}\,dx$$ for $a,b>0$? My methods: Let $a,b > 0$ and let $$I(b)=\int_{0}^{+\infty}e^{\left(-ax^2-\frac ...
3
votes
2answers
357 views

To determine whether the integral $\int_0^{\infty} \frac{\sin{(ax+b)}}{x^p} \,\mathrm dx$ converges for $p>0$

If $p >0$, determine if the following integral converges: $$\int_1^{\infty} \frac{\sin{(ax+b)}}{x^p} \,\mathrm dx$$ What i did so far is: If $p>1$, then $$f(x)= ...
8
votes
2answers
237 views

Compute $\int_0^1 \frac{\arcsin(x)}{x}dx$

$$\int_0^1 \frac{\arcsin(x)}{x}dx$$ This is a proposed for a Calculus II exam, and I have absolutely no idea how to solve it. Tried using Frullani or Lobachevsky integrals, or beta and gamma ...
5
votes
3answers
203 views

Applications of the Residue Theorem to the Evaluation of Integrals and Sums

Evaluate the integral $$\int_{-\infty}^{\infty} \frac{1}{(1 + x^2)^{n+1}} dx. $$ I know that it equals $2\pi i$(the sum of the residues; at $z_k$) where $z_k$ are the poles of the function. I ...