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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2
votes
0answers
32 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 ...
1
vote
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 ...
-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. ...
2
votes
0answers
58 views

Double integral in complex variables form. [closed]

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 $y$ ...
0
votes
2answers
44 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 ...
0
votes
1answer
49 views
+50

Integrating an expression wrt to a variable which is a function of variables which appear in the expression?

Say I have an expression like this: $$\partial C/\partial a = \frac{(a - y)}{a(1-a)x}$$ where $a$ and $y$ are independent, but $a$ is a function of $x$ and possibly some other variables (i.e. $a = f(...
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.
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-...
5
votes
0answers
82 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 $\...
0
votes
2answers
44 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 ...
3
votes
1answer
66 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 ...
-2
votes
2answers
73 views

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

What is the indefinite integral of $\frac{4^x}{e^x}$? Can anyone can show a step by step solution for this problem?
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
0answers
38 views
+150

Integral involving the von Mises-Fisher distribution

I'm going quickly through the VonMises-Fisher distribution $M$ on $\mathbb S^{d-1}$ and its properties. Its probability density function is: $$f(x; \kappa,\mu)= c(\kappa)\exp(\kappa x^T\mu)$$ where $...
13
votes
5answers
203 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 $y=x^2$)...
0
votes
0answers
42 views

Work required to align pieces in a plane.

Given two piecewise continuous functions f(x) and g(x) and that $\lim_{a -> x^-} g(a) - f(a) = \lim_{a -> x^+} g(a) - f(a)$ at all points, find the work used to shift each of the planar slolids ...
1
vote
1answer
34 views

Is integration with respect to spherical measure equivalent to manifold integration over sphere?

Let $S$ be an $n$--sphere $\mathcal{S}^n(0,R)\subset\mathbb{R}^{n+1}$, $\Theta\subset\mathbb{R}^n$ an open subset and let $\phi:\Theta\subset\mathbb{R}^n\longrightarrow S\subset\mathbb{R}^{n+1}$ be a ...
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. ...
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
78 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 ...
5
votes
5answers
112 views

Bounding a series: $\frac{\pi}{2} < \sum_{n=0}^\infty \frac{1}{n^2 + 1} < \frac{3\pi}{2} $

I have the following statement - $$\frac{\pi}{2} < \sum_{n=0}^\infty \dfrac{1}{n^2 + 1} < \frac{3\pi}{2} $$ So I tried to prove this statement using the integral test and successfully proved ...
1
vote
2answers
41 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 ...
0
votes
3answers
68 views

Compute $\mathbb{P}(1<X^2+Y^2<2)$ when $(X,Y)$ is i.i.d. standard normal

Assume that $(X,Y)$ is i.i.d. standard normal. Compute $\mathbb{P}(1<X^2+Y^2<2)$. So I've decided to use polar coordinates to solve and I've gotten to this point: $$\iint_{1\lt X^2+Y^2\lt2} ...
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
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 ...
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 ...
0
votes
2answers
69 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
182 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: $$\int \arctan(\sqrt{x+2})dx = \int \arctan(\sqrt{x+2}) \cdot 1\ dx$$ ...
0
votes
2answers
78 views

Integration by parts proof 1 = 0

Let's integrate $\int\frac{f^\prime(x)}{f(x)} dx$ by parts $$ \\ \mbox{ Let } dv= f^\prime(x)dx,u=\frac{1}{f(x)} \\ \mbox{ Then }v=f(x), du=-\frac{f^\prime(x)}{[f(x)]^2}dx \\ \mbox{ This implies }\int\...
0
votes
0answers
10 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)...
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 ...
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 ...
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 ...
-1
votes
0answers
70 views

Evaluating double integral [closed]

I couldn't get it Someone can help me about this? Thanks a lot.
-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?
0
votes
1answer
20 views

Short-time Fourier Transform identity in $L^2$

Define the Short-time (or windowed) Fourier Transform of a function $f:\mathbb{R}\rightarrow\mathbb{C}$ as follows, $F_gf(\omega,t)=\int\limits_{\mathbb{R}}f(x)\overline{g(x-t)e^{ix\omega}}dx$. Show ...
0
votes
3answers
85 views

Evaluate $\iiint dx\,dy\,dz$ betweem $x=0,y=0,z=0, x+y+2z=2$

I need to evaluate $$\iiint dx\,dy\,dz$$ the volume between $x=0,y=0,z=0, x+y+2z=2$ I am stuck about choosing the limits of integration, I think that the limits should be: $$\int_0^{(2-x-y)/2}\...
1
vote
0answers
52 views

Find the Log and dfrac integral closed form? [closed]

Find the $$f(a)=\int_{0}^{+\infty}\dfrac{\ln^2{\left(\cot{\left(\dfrac{ax}{2}-\frac{\pi}{4}\right)}\right)}}{x^4+4}dx$$ where $a>0$ Try this wwo classic methods $x\to\dfrac{1}{x}?$ or $f'(a)$ ...
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 ...
2
votes
1answer
73 views

Why are functions that are continuous over $[a,b]$ integrable over $[a,b]$?

Why are functions that are continuous over $[a,b]$ integrable over $[a,b]$? Why is it that to be Riemann-integrable the infimum of the upper sums and the supremum of the lower sums have to be equal? ...
0
votes
0answers
23 views

Parametric functions to describe the intersection of two orthogonal cylinder surfaces?

I am trying to find a parametric equation for the intersection line of the surface of two orthogonal cylinders, $\vec{P}$ is a point that belongs to this intersection: $$\vec{P(t)} = \begin{bmatrix} ...
-1
votes
0answers
14 views

Is there any equality for the integral of the product of normal derivative?

I am trying to get the proof of $\int\int_DD_uf(x) D_ug(x) dx$. For example in Green Theorem, in integral we use the product of $ \nabla$, when it comes to normal derivative, how can I organize the ...
1
vote
0answers
27 views

Calculate the curvilinear integral

I need to calculate the curve integral. This should be the curve integral of I rank, which can be calculated with the formula : $$\int_{C}f(x,y)ds=\int_{a}^{b}f(g(t),h(t)) \sqrt{(\frac{dx}{dt})^2+\...
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
2answers
117 views

Evaluation of Irrational Integral

Evaluation of $$\int\frac{x^4}{(1-x^4)^{\frac{3}{2}}}dx$$ $\bf{My\; Try::}$ Let $$I = \int\frac{x^4}{(1-x^4)^{\frac{3}{2}}}dx = -\frac{1}{4}\int x\cdot \frac{-4x^3}{(1-x^{4})^{\frac{3}{2}}}dx$$ ...
0
votes
4answers
105 views

Value of $\int\tan^{-1}(x)\,dx$

What is the value of $\int^{1000}_{0}\tan^{-1}(x)\,\mathrm d x$? Today we were taught about graphs of all trigonometric inverse functions. So my proofessor split it into $0-\tan(1)$ and $\tan(1)-...
0
votes
4answers
126 views

Strange integral result

Consider the following integral, $$\mathrm{I} = \int_{-1}^{1}\frac{d}{dx}\tan^{-1}\left(\frac{1}{x}\right)dx$$ We can do this in two ways, First Using the fact that the antiderivative of $\frac{d}{...