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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2
votes
1answer
126 views

How do you integrate Gaussian integral with contour integration method?

How do you integrate $$\int^{\infty}_{-\infty} e^{-x^2} dx$$ with contour integration method? I do not even know how to setup the problem.
2
votes
1answer
124 views

Show that $\int^{\infty}_0 \frac{\sin^4 x}{x^4}=\frac{\pi} 3$. [duplicate]

Show that $$\int^{\infty}_0 \left( \frac{\sin x}{x}\right)^4=\frac{\pi} 3$$ Although I know the integral with the index is $1$ and $2$, I have no idea on this one. Please help.
9
votes
3answers
280 views

A probabilistic integral $\int_{-\infty}^{\infty}e^{-x^2/2\sigma^2}\arcsin\left(1-2\left|\lfloor x\rceil-x\right|\right)\,dx$

In my probabilistic studies, a tough integral appeared. Note that $\lfloor x\rceil$ is not the floor function; it is the nearest integer function. Up to some constants, it appears in a Buffon-like ...
3
votes
0answers
101 views

Help on the Integration of $\int_0^{\infty} e^{-bx}\sin ax^2 \, \mathrm{d}x$.

I have had the misfortune of coming across the following integral, for real $b$ and $a > 0$: $$\int\limits_{0}^{\infty} e^{-bx} \sin\left(ax^{2}\right) \, \mathrm{d}x.\tag{1}$$ Naturally, I ...
2
votes
3answers
95 views

Evaluate $\int_{1}^{\infty} \frac{\ln{(2x-1)}}{x^2} $

$$\int_{1}^{\infty} \frac{\ln{(2x-1)}}{x^2} dx$$ My approach is to calc $$\int_{1}^{X} \frac{\ln{(2x-1)}}{x^2} dx$$ and then take the limit for the answer when $X \rightarrow \infty$ However, I must ...
-3
votes
1answer
40 views

How to find the integral of $\int \frac{GMm}{r^2}\,dr$ [closed]

I want to find the integral of: $$\int_R^\infty \frac{GMm}{r^2}\,dr$$
1
vote
4answers
70 views

Integration of $x/(x^2+1)$ from $-\infty$ to $\infty$

I am trying to find the area of this graph $\int_{-\infty}^\infty\frac{x}{x^2 + 1}$ The question first asks to use the u-substitution method to calculate the integral incorrectly by evaluating ...
3
votes
2answers
57 views

Improper integral of a cosine

I'm trying to follow some equations in an electrical engineering paper that I'm reading. I'll spare you the details, but at one point I come across: $$\lim_{ T \rightarrow \infty }\int_{-T/2}^{T/2} ...
-1
votes
1answer
83 views

Evaluate $\int_0^{+\infty } \frac{\log(t)}{1+t^2} \, dt$ [duplicate]

How can we compute $$I=\int_0^{+\infty } \frac{\log(t)}{1+t^2} \, dt$$ Mathematica gives $I=0$.
4
votes
3answers
275 views

Prove $\int_{\mathbb{R^{+}}} \frac{\sin^3 {(\pi x^2)} \cos {(4x^2)}}{x^5} dx=\frac{\pi}{32} (3\pi-4)^2$

How do you arrive at the result $$I=\displaystyle\int_{\mathbb{R^{+}}} \dfrac{\sin^3 {(\pi x^2)} \cos {(4x^2)}}{x^5} dx=\dfrac{\pi}{32} (3\pi-4)^2\ ?$$ Wolfram Alpha agrees numerically. I tried ...
2
votes
2answers
94 views

How to do integral $\int_0^{\infty} e^{-x^2-ax^4}\ dx , \ \text{ for $a>0$}$

I was told by this OP, $$\int_{0}^{\infty} e^{\large-x^n} \,dx =\Gamma \left(\frac{n+1}{n}\right), \qquad\text{ for $n>1$}.$$ This is via the variable change $t=x^n$: $$\int_{0}^{\infty} ...
0
votes
2answers
129 views

Improper integral of $\frac{\ln x}x$

Find $$\int_e^{\infty}\frac{\ln x}{x}\ dx$$ $A.\ \dfrac12$ $B.\ \dfrac{e^2}{2}$ $C.\ \dfrac{\ln(2e)}{2}$ $D.$ DNE (Does not exist) I tried doing this and this is where I've gone so far: $$\lim ...
1
vote
3answers
130 views

Evaluate $\int_0^\infty\frac{dl}{(r^2+l^2)^{\frac32}}$

How to evaluate the following integral $$\int_0^\infty\frac{dl}{(r^2+l^2)^{\large\frac32}}$$ The solution is supposed to look like this, unfortunately I can't derive it. $$ ...
2
votes
5answers
144 views

Show that $\int^\infty_0\left(\frac{\ln(1+x)} x\right)^2dx$ converge.

Show that $$\int\limits^\infty_0\left(\frac{\ln(1+x)} x\right)^2dx$$ converge. I have utterly no clue on this integral. Please give me some hints. Thanks you.
1
vote
3answers
134 views

Value of the integral $\int_{\mathbb{R}} \frac{x\sin {(\pi x)}}{(1+x^2)^2}$

How do we evaluate the integral $$I=\displaystyle\int_{\mathbb{R}} \dfrac{x\sin {(\pi x)}}{(1+x^2)^2}$$ I have wasted so much time on this integral, tried many substitutions $(x^2=t, \ \pi x^2=t)$. ...
1
vote
0answers
39 views

Partial fraction for integrating

I have been trying to solve the integral $\displaystyle\int \frac{dx}{(x-1)^2 (x^2+1)^3}$. So while trying to get the partial fraction which way is better? ...
9
votes
2answers
177 views

An exercise from my brother: $\int_{-1}^1\frac{\ln (2x-1)}{\sqrt[\large 6]{x(1-x)(1-2x)^4}}\,dx$

My brother asked me to calculate the following integral before we had dinner and I have been working to calculate it since then ($\pm\, 4$ hours). He said, it has a beautiful closed form but I doubt ...
14
votes
1answer
261 views

Prove ${\large\int}_0^\infty\left({_2F_1}\left(\frac16,\frac12;\frac13;-x\right)\right)^{12}dx\stackrel{\color{#808080}?}=\frac{80663}{153090}$

I discovered the following conjectured identity numerically (it holds with at least $1000$ digits of precision). How can I prove it? ...
4
votes
3answers
154 views

Evaluate $\int_0^1\frac{x^a-x^{-a}}{x-1}dx$

I have heard that: $$\int_0^1\frac{x^a-x^{-a}}{x-1}dx=\frac1 a-\pi\cot(\pi a)$$ when $-1<a<1$. How would I prove this? That doesn't have an elementary indefinite integral, but the definite ...
2
votes
2answers
61 views

Duo Fresnel-like integrals $(??)$

I really wonder how I can prove the following integrals. $$\int_0^\infty \sin ax^2\cos 2bx\, dx=\frac{1}{2}\sqrt{\frac{\pi}{2a}}\left(\cos \frac{b^2}{a}-\sin\frac{b^2}{a}\right)$$ and ...
1
vote
1answer
70 views

Explain why $\big(\int_{-\infty}^{\infty}e^{-z^2/2}dz \big)^2 = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(z^2 + u^2)/2}dzdu$

I came across the following when studying a proof related to the normal distribution: $$\left(\int_{-\infty}^{\infty}e^{-z^2/2}\ dz \right)^2 = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(z^2 ...
4
votes
2answers
88 views

Evaluate $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-\frac{1}{2}(x^2-xy+y^2)}dx\, dy$

I need to evaluate the following integral: $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-\frac{1}{2}(x^2-xy+y^2)}dx\, dy$$ I thought of evaluating the iterated integral ...
20
votes
1answer
343 views

Prove ${\large\int}_0^\infty\frac{\ln x}{\sqrt{x}\ \sqrt{x+1}\ \sqrt{2x+1}}dx\stackrel?=\frac{\pi^{3/2}\,\ln2}{2^{3/2}\Gamma^2\left(\tfrac34\right)}$

I discovered the following conjecture by evaluating the integral numerically and then using some inverse symbolic calculation methods to find a possible closed form: $$\int_0^\infty\frac{\ln ...
21
votes
3answers
537 views
+500

Integral ${\large\int}_0^\infty\frac{\ln x}{1+x}\sqrt{\frac{x+\sqrt{1+x^2}}{1+x^2}}\ \mathrm dx$

Please help me to evaluate this integral: $$ I={\large\int}_{0}^{\infty}{\ln\left(x\right) \over 1 + x}\, \,\sqrt{\,x + \sqrt{\,1 + x^{2}\,}\, \over 1 + x^{2}\,}\,\,{\rm d}x.\tag1 $$ Mathematica could ...
9
votes
2answers
230 views

Evaluate $\int_{0}^{\large\frac{\pi}{4}} \ln {(\sin x)}\cdot\ln {(\cos x)} \left(\frac{\ln{(\sin x)}}{\cot x}+\frac{\ln {(\cos x)}}{\tan x}\right)dx$

How do I find the value of this integral? $$I=\int_{0}^{\Large\frac{\pi}{4}} \ln {(\sin x)}\cdot\ln {(\cos x)} \left(\dfrac{\ln{(\sin x)}}{\cot x}+\dfrac{\ln {(\cos x)}}{\tan x}\right)dx$$ I tried ...
4
votes
1answer
50 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
47 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,\\ ...
10
votes
2answers
302 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 ...
0
votes
2answers
43 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
50 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: ...
2
votes
1answer
91 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 ...
2
votes
2answers
40 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.
5
votes
3answers
121 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 ...
6
votes
5answers
212 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 ...
1
vote
1answer
97 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 ...
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 ...
2
votes
2answers
40 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 ...
1
vote
0answers
18 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 ...
10
votes
3answers
192 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 ...
3
votes
0answers
31 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 $(a, b)$, ...
3
votes
2answers
128 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 ...
1
vote
3answers
32 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 }_{ ...
0
votes
1answer
32 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 ...
3
votes
1answer
38 views

Show $\iint xye^{-xy}\,dx\,dy$ is convergent or divergent

Determine convergence/divergence of $$\iint xye^{-xy}\,dx\,dy$$ for $x,y \geqslant 0$ i.e. in the first quadrant. I have managed to show that $xye^{-xy} \to 0$ in the first quadrant but other ...
4
votes
1answer
78 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 ...
1
vote
1answer
63 views

Help finishing this exercise!

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 $F$ is ...
3
votes
1answer
41 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 ...
3
votes
1answer
45 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 ...
1
vote
0answers
154 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 ...