Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of ...

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0
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0answers
55 views

Solve integral : $\int_c^d \int _a^b \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{1}{2\sigma^2}(x - \mu)^2\right) \, d\sigma d\mu$

I'm trying to compute the integral \begin{align*} \int_c^d \int _a^b \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{1}{2\sigma^2}(x - \mu)^2\right) \, d\sigma d\mu, \end{align*} where $a<b$ and $c&...
0
votes
5answers
80 views

How do you solve trig integrals using recursions?

My calculus professor gave us the problem $\int \sin^2(x)\cos^2(x)dx$ and told us to solve it via recursion but I can't seem to find how to do it in my textbook.
2
votes
0answers
23 views

Splitting the region and estimating fractional Sobolev norms

I've been reading the paper "On the Bourgain, Brezis, and Mironescu Theorem Concerning Limiting Embeddings of Fractional Sobolev Spaces" by Maz'ya and Shaposhnikova and struggling with the short style ...
3
votes
1answer
82 views
+50

Drawing large rectangle under concave curve

Let $f$ be a continuous concave function on $[0,1]$ with $f(1)=0$ and $f(0)=1$. Does there exist a constant $k$ for which we can always draw a rectangle with area at least $k\cdot \int_0^1f(x)dx$, ...
3
votes
4answers
116 views
1
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0answers
26 views

The asymptotic behaviour of $\sum_{1\leq k\leq N-1}\int_{p_k}^{p_{k+1}}\log x d[x]$, where $p_n$ is the nth prime number

Let $p_k$ is the kth prime number and consider for $N\geq 2$ the arithmetic function $$f(N)=\sum_{k=1}^{N-1}\int_{p_k}^{p_{k+1}}\log(x) d[x]$$ where $[x]$ is the integer part function (provide us in ...
2
votes
0answers
58 views

Double integral in complex variables form. [on hold]

Rewrite $\displaystyle \iint f(x,y) dx \, dy$ in complex variable form of $\displaystyle \iint g(z, \bar{z}) dz \, d\bar{z}$? Where $z=x+iy$, $\bar{z}=x-iy$ and $x$ changes from $0$ to $a$ and $y$ ...
-1
votes
0answers
30 views

Romberg, trapezoidal rule exact for polynomials

My question is, how can I proof that the rombergs method of the summed trapezoidal rule is exact for polynomials with degree $(2n+1)$ or less. Thanks for helping, one or two tips can help me here. ...
0
votes
2answers
42 views

Integrating an infinite-valued function over a zero length interval

Let $\delta(t)$ be defined as the limit of a Gaussian pdf with 'zero variance'. What is then the result of $$I=\int_0^0 \delta(t)dt\quad?$$ on the one hand, "$\delta(0)=\infty$", but the length of ...
1
vote
1answer
40 views

line integral, stuck in the integral step…

Problem: A uniform wire has the shape of that portion of the curve of intersection of the two surfaces $x^2+y^2=z^2$ and $y^2=x$ connecting the points $(0,0,0)$ and $(1,1,\sqrt{2})$. Find the z-...
13
votes
5answers
192 views

I want to show that $\int_{-\infty}^{\infty}{\left(x^2-x+\pi\over x^4-x^2+1\right)^2}dx=\pi+\pi^2+\pi^3$

I want to show that $$\int_{-\infty}^{\infty}{\left(x^2-x+\pi\over x^4-x^2+1\right)^2}dx=\pi+\pi^2+\pi^3$$ Expand $(x^4-x+\pi)^2=x^4-2x^3+2x^2-2x\pi+\pi{x^2}+\pi^2$ Let see (substitution of $y=x^2$)...
5
votes
0answers
76 views

Integral with an infinite sum

Let: $$\mathcal{S}(x)=\sum_{k=1}^{\infty}-\frac{\cos(\frac{\pi}{2}+k x)}{k^x}$$ I need help evaluating $$\int_{0}^{1}\mathcal{S}(x) dx$$ Obviously the cosine term in the numerator simplifies to $\...
3
votes
1answer
64 views

How to evaluate this Fourier Transform $A\int_{-\infty}^{\infty} \frac{e^{ikx}}{(1+x^{2})^{\frac{\nu+1}{2}}}dx$

This is basically the Fourier transform of a Student´s T pdf. How do we compute it? $$A\int_{-\infty}^{\infty} \frac{e^{ikx}}{(1+x^{2})^{\frac{\nu+1}{2}}}dx$$ for $\nu$ any number greater than zero ...
0
votes
2answers
43 views

Antidifferentiation: Stone dropped from $150ft$ rising at $10ft/sec$

A stone is dropped from a balloon when it is $150ft$ above the ground and rising at the rate of $10ft/sec$. How long will it take the stone to strike the ground, and with what velocity does it strike ...
-2
votes
2answers
73 views

What is the indefinite integral of $\frac{4^x}{e^x}$? [on hold]

What is the indefinite integral of $\frac{4^x}{e^x}$? Can anyone can show a step by step solution for this problem?
0
votes
3answers
2k views

Volume of a horizontal cylinder using height of liquid

“Tanks” are cylinders with circular cross-section and axis horizontal. These cylinders are variable in size with radius and length different for each tank. We need to determine the amount of liquid ...
2
votes
4answers
50 views

Antiderrivative of ${d^2 y \over dx^2} = 1-x^2$

At any point $(x,y)$ on a curve, ${d^2 y \over dx^2} = 1-x^2$, and an equation of the tangent line to the curve at the point $(1,1)$ is $y=2-x$. Find an equation of the curve. This is what I've done ...
2
votes
1answer
76 views

Compute the probability of a joint event involving two independent standard normals

Suppose $X$ and $Y$ are independent, standard normal random variables. I'm trying to compute the probability of the event $$ \{X \leq x, Y \leq kX\} $$ where $k$ is a positive constant. The ...
14
votes
3answers
566 views

How to integrate $\int_{0}^{\infty }{\frac{\sin x}{\cosh x+\cos x}\cdot \frac{{{x}^{n}}}{n!}\ \text{d}x} $?

I have done one with $\displaystyle\int_0^{\infty}\frac{x-\sin x}{x^3}\ \text{d}x$, but I have no ideas with these: $$\begin{align*} I&=\int_{0}^{\infty }{\frac{\sin x}{\cosh x+\cos x}\cdot \frac{{...
5
votes
2answers
100 views

Complicated Laplace Transform

I have found the following Laplace Transform in a list $$\int\limits_0^{\infty}e^{-st}\frac{e^{-u^2/4t}}{\sqrt{\pi t}}dt = \frac{e^{-u\sqrt{s}}}{\sqrt{s}}.$$ I am wondering how to prove this? I ...
1
vote
4answers
130 views

Evaluate $\int_{-\pi}^{\pi} x^2 \cos{3x}dx$

Evaluate $$\int_{-\pi}^{\pi} x^2 \cos{3x}dx$$ I applied integration-by-parts twice and finally got a result of $-\frac{4\pi}{9}$ but the back of the book says $+\frac{4\pi}{9}$ . Which answer is ...
1
vote
1answer
49 views

Calculate an integral with limit another integral

I have a list of integrals to do with a structure similar to this one, but I don't know how to attack anyone of them. I hope you can help me doing this one to understand how to do the other ones. ...
1
vote
1answer
77 views

Partial fraction integration problem

I'm trying to solve this integral by partial fraction: $$ \int \frac{2x-6} {(x-2)^2(x^2+4)} dx \ $$ i think i have to write the expression like $$ 2\int \frac{x-3} {(x-2)^3(x+2)} dx \ $$ Then i ...
1
vote
1answer
30 views

Volume by integration - Disk Method only/Non-coordinate axis

PROBLEM: Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line x = 5. (Use disk method) $$ xy = 3, y = 1, y = 4, x = 5 $$ So first I ...
2
votes
0answers
106 views

Evaluating an integral in 3 different intervals [closed]

Evaluate $$\int_{-5}^{-2}\Big(\frac{x^2-x}{x^2-3x+1}\Big)^2dx\\+ \int_{1/6}^{1/3}\Big(\frac{x^2-x}{x^2-3x+1}\Big)^2dx\\+ \int_{6/5}^{3/2}\Big(\frac{x^2-x}{x^2-3x+1}\Big)^2dx\\$$ I don't think ...
16
votes
3answers
338 views

Prove $\pi^2\int_0^\infty\frac{x\sin^4\pi x}{\cos\pi x+\cosh\pi x}dx=e^2\int_0^\infty\frac{x\sin^4ex}{\cos ex+\cosh ex}dx=\frac{176}{225}$

Marco Cantarini and Jack D'Aurizio proved hard-looking integrals (see Marco and Jack) in my recent two posts. This is our final hard-looking integral that yield a rational answer: $$\pi^2\int_{...
21
votes
2answers
717 views

Why does the hard-looking integral $\int_{0}^{\infty}\frac{x\sin^2(x)}{\cosh(x)+\cos(x)}dx=1$?

I have to ask this question; most looking complicated definite integral yield not so nice closed form or irrational numbers or mixed of what ever ect. Why is this particular hard looking integral ...
1
vote
2answers
40 views

Finding the shap of the volume $\int_0^{\pi/2}\int_0^{\pi/2}\int_0^{1} \left(\rho^2 \sin \phi \right) d \rho d \phi d \theta$

I need to find the shap of the volume:$$\int_0^{\pi/2}\int_0^{\pi/2}\int_0^{1} \left(\rho^2 \sin \phi \right) \,\mathrm d \rho \,\mathrm d \phi\,\mathrm d \theta$$ I thought that the shape is ...
3
votes
2answers
75 views

Show that $\int\limits_a^b |f(t)|dt \leq (b-a)\int\limits_a^b|f'(t)|dt$

Let $f:[a,b]\to\mathbb{R}$ be continuously differentiable. Suppose $f(a) = 0$. Show that $$ \int\limits_a^b|f(t)|dt \leq (b-a)\int\limits_a^b|f'(t)|dt $$ By the mean value theorem, for every $t\in[...
0
votes
2answers
68 views

Integrate logarithmic derivative of a periodic function

Given $f$ a $p$-periodic function over $\mathbb{C}$, how to show that : $$\frac{1}{\mathrm{i}p}\int_a^{a+p}\frac{f'(t)}{f(t)}dt \in \mathbb{Z}$$ Is there any elegant method ?
1
vote
6answers
180 views

Solve definite integral: $\int_{-1}^{1}\arctan(\sqrt{x+2})\ dx$

I need to solve: $$\int_{-1}^{1}\arctan(\sqrt{x+2})\ dx$$ Here is my steps, first of all consider just the indefinite integral: $$\int \arctan(\sqrt{x+2})dx = \int \arctan(\sqrt{x+2}) \cdot 1\ dx$$ ...
2
votes
1answer
40 views

Where is the mistake of a possible application of Frullani's theorem in this case?

My question is about what is the problem, if there is one, to get an identity using Frullani's integral. I've in a hand the statement from MathWorld, and other statement from this site, with a nice ...
1
vote
1answer
35 views

Definite integral involving algebraic, exponential, and product of two Meijer's G function

I am having trouble with calculating the following integral: \begin{equation} I = \int_{0}^{\infty}x\exp({-\beta x})\large{G}_{2,2}^{1,2}\left( x \left| \begin{array}{cc} 1,1 \\ 1,0 \end{array} \...
0
votes
2answers
2k views

Laplace Transform of $tf(t)$

Q. prove that $\mathfrak{L}\{tf(x)\}=-\frac{d\mathfrak{L}\{f(x)\}}{ds}$ where the notation used is standard one. Attempt I tried what would seem obvious way to start: $$\mathfrak{L}\{tf(x)\}=\int_0^\...
0
votes
0answers
48 views

Unusual integration of 1/cx [duplicate]

Consider an integral: $$\int_2^3 \frac{1}{cx} dx$$ where $c$ is a constant So we can take that out of the integral, so $$\int_2^3 \frac{1}{cx} dx = \frac{1}{c} \int_2^3 \frac{1}{x} dx $$ all is ...
1
vote
1answer
90 views

A two-dimensional integral related to a Gaussian distribution

I am trying to evaluate the integral $I=\int_a^b\int_a^b\frac{1}{\sqrt{2\pi}\theta}e^{-\frac{(x-y)^2}{2\theta^2}}dxdy$. With the aid of Mathematica software, the result is $I=\left(e^{-\frac{(a-b)^...
-1
votes
0answers
70 views

Evaluating double integral [closed]

I couldn't get it Someone can help me about this? Thanks a lot.
3
votes
1answer
98 views

A puzzle about integrability

I know there is a Proposition: for $f(x)$ is bounded on $[a,b]$,then $f(x)$ is integrable if and only if given $\epsilon>0$ ,there exists a partition such that $U(f,P)-L(f,P)<\epsilon$ But my ...
2
votes
4answers
246 views

How to find the area enclosed by the ellipse $b^2x^2 + a^2y^2=a^2b^2$?

I need to find the area enclosed by the ellipse $b^2x^2 + a^2y^2=a^2b^2$, and I know it involves taking the integral, but I'm not sure what function I should be taking the integral of or how to find ...
2
votes
2answers
45 views

Computing a double integral with applications to prime numbers

I was reading the preprint [1] which contains on p. 7 the following formula (for $4<s\le6$): $$ f_1(s)=\frac{2e^\gamma}{s}\left\{\log(s-1)+\int_4^s\int_3^t\frac{\log(u-2)}{u-1}du\,dt \right\} $$ ...
0
votes
1answer
36 views

Calculate the volume of a body bounded by planes, using double integral [on hold]

I try to calculate volume in the first octant bounded by the coordinate planes, the plane $y=4$ and the plane $$ \left ( \frac{x}{3} \right )+\left ( \frac{z}{5} \right )=1. $$ Can someone help me? I ...
0
votes
0answers
9 views

Bounds on flux integrals

What are some handy upper bounds for surface integrals (and their proofs)? Specifically, suppose $f$ is a bounded function on a surface $S$. Do we have $$ \int_{\partial S} F \cdot n \; \mathrm{d}S \...
1
vote
0answers
58 views

Limit of $\sum_{k=0}^{n}\frac{1}{2k+n}$ and similar

Examine wether following sequences have limits and if yes - find them. a)$\sum_{k=0}^{n}\frac{1}{2k+n}$ b)$\sum_{k=0}^{n}\frac{(-1)^n}{2k+n}$ c)$\sum_{k=0}^{n}\frac{(-1)^k}{2k+n}(\frac{1}{3})^k$ a)...
5
votes
5answers
527 views

Evaluate $\int_0^{{\pi}/{2}} \log(1+\cos x)\, dx$

Find the value of $\displaystyle \int_0^{{\pi}/{2}} \log(1+\cos x)\ dx$ I tried to put $1+ \cos x = 2 \cos^2 \frac{x}{2} $, but I am unable to proceed further. I think the following integral can be ...
10
votes
4answers
255 views

Evaluating $\int_{0}^{\pi}\ln (1+\cos x)\, dx$ [duplicate]

The problem is $$\int_{0}^{\pi}\ln (1+\cos x)\ dx$$ What I tried was using standard limit formulas like changing $x$ to $\pi - x$ and I also tried integration by parts on it to no avail. Please help....
1
vote
1answer
29 views

Integration of periodic function $f \in L^1([0, 2\pi])$

While studying trigonometric series and $L^p$ spaces I was wondering the following: Let's say we have a $2\pi$-periodic function $f \in L^1([0, 2\pi])$ satisfying $\int_{0}^{2\pi}f(x) \, dx = 0$. Is ...
4
votes
0answers
495 views

Integral of two error functions (erf)

In my research I came across the following integral: \begin{equation} \int_{-\infty}^{+\infty}\frac{\partial{p(t)}}{\partial{t}}\frac{1}{4}\Big(1-\operatorname{erf}\Big(\frac{t-a}{\sigma\sqrt{2}}\Big)\...
-2
votes
2answers
63 views

Double Integral questions [closed]

It was my exam question about double integral and I couldn't do it. But I wanna learn this because it is important. Can you help me?
4
votes
3answers
112 views

Find all continuous functions $f:[0,1]\rightarrow \mathbb{R}$ that satisfy: $\int_0^1 f(x)dx=1/3 + \int_0^1 f^2(x^2)dx$

(Note that $f^2(x)=f(x)\cdot f(x)$ and not composition.) Since both integrals are defined, derivation is out of the question. I tried integrating the second integral by parts but reached something ...