# Tagged Questions

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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### 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 ...
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### 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 ...
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### 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. ...
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### 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 $y$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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 $... 5answers 203 views ### I want to show that$\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$y=x^2)... 0answers 42 views ### Work required to align pieces in a plane. Given two piecewise continuous functions f(x) and g(x) and that\lim_{a -> x^-} g(a) - f(a) = \lim_{a -> x^+} g(a) - f(a)$at all points, find the work used to shift each of the planar slolids ... 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 ... 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. ... 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 ... 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 ... 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 ... 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 ... 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 ... 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: $$\iint_{1\lt X^2+Y^2\lt2} ... 2answers 75 views ### Show that \int\limits_a^b |f(t)|dt \leq (b-a)\int\limits_a^b|f'(t)|dt Let f:[a,b]\to\mathbb{R} be continuously differentiable. Suppose f(a) = 0. Show that$$ \int\limits_a^b|f(t)|dt \leq (b-a)\int\limits_a^b|f'(t)|dt $$By the mean value theorem, for every t\in[... 0answers 48 views ### Unusual integration of 1/cx [duplicate] Consider an integral:$$\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 $$all is ... 1answer 40 views ### Where is the mistake of a possible application of Frullani's theorem in this case? My question is about what is the problem, if there is one, to get an identity using Frullani's integral. I've in a hand the statement from MathWorld, and other statement from this site, with a nice ... 2answers 69 views ### Integrate logarithmic derivative of a periodic function Given f a p-periodic function over \mathbb{C}, how to show that :$$\frac{1}{\mathrm{i}p}\int_a^{a+p}\frac{f'(t)}{f(t)}dt \in \mathbb{Z}$$Is there any elegant method ? 6answers 182 views ### Solve definite integral: \int_{-1}^{1}\arctan(\sqrt{x+2})\ dx I need to solve:$$\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$$... 2answers 78 views ### Integration by parts proof 1 = 0 Let's integrate \int\frac{f^\prime(x)}{f(x)} dx by parts$$ \\ \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\... 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 $$\int_{\partial S} F \cdot n \; \mathrm{d}S \... 0answers 58 views ### Limit of \sum_{k=0}^{n}\frac{1}{2k+n} and similar Examine wether following sequences have limits and if yes - find them. a)\sum_{k=0}^{n}\frac{1}{2k+n} b)\sum_{k=0}^{n}\frac{(-1)^n}{2k+n} c)\sum_{k=0}^{n}\frac{(-1)^k}{2k+n}(\frac{1}{3})^k a)... 2answers 45 views ### Computing a double integral with applications to prime numbers I was reading the preprint [1] which contains on p. 7 the following formula (for 4<s\le6):$$ 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\} $$... 1answer 36 views ### Calculate the volume of a body bounded by planes, using double integral [on hold] I try to calculate volume in the first octant bounded by the coordinate planes, the plane y=4 and the plane$$ \left ( \frac{x}{3} \right )+\left ( \frac{z}{5} \right )=1. $$Can someone help me? I ... 1answer 30 views ### Volume by integration - Disk Method only/Non-coordinate axis PROBLEM: Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line x = 5. (Use disk method)$$ xy = 3, y = 1, y = 4, x = 5 $$So first I ... 1answer 29 views ### Integration of periodic function f \in L^1([0, 2\pi]) While studying trigonometric series and L^p spaces I was wondering the following: Let's say we have a 2\pi-periodic function f \in L^1([0, 2\pi]) satisfying \int_{0}^{2\pi}f(x) \, dx = 0. Is ... 0answers 70 views ### Evaluating double integral [closed] I couldn't get it Someone can help me about this? Thanks a lot. 2answers 63 views ### Double Integral questions [closed] It was my exam question about double integral and I couldn't do it. But I wanna learn this because it is important. Can you help me? 1answer 20 views ### Short-time Fourier Transform identity in L^2 Define the Short-time (or windowed) Fourier Transform of a function f:\mathbb{R}\rightarrow\mathbb{C} as follows, F_gf(\omega,t)=\int\limits_{\mathbb{R}}f(x)\overline{g(x-t)e^{ix\omega}}dx. Show ... 3answers 85 views ### Evaluate \iiint dx\,dy\,dz betweem x=0,y=0,z=0, x+y+2z=2 I need to evaluate$$\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}\... 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$f'(a)$... 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 ... 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? ... 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: $$\vec{P(t)} = \begin{bmatrix} ... 0answers 14 views ### Is there any equality for the integral of the product of normal derivative? I am trying to get the proof of \int\int_DD_uf(x) D_ug(x) dx. For example in Green Theorem, in integral we use the product of \nabla, when it comes to normal derivative, how can I organize the ... 0answers 27 views ### Calculate the curvilinear integral I need to calculate the curve integral. This should be the curve integral of I rank, which can be calculated with the formula :$$\int_{C}f(x,y)ds=\int_{a}^{b}f(g(t),h(t)) \sqrt{(\frac{dx}{dt})^2+\... 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 ... 2answers 117 views ### Evaluation of Irrational Integral Evaluation of $$\int\frac{x^4}{(1-x^4)^{\frac{3}{2}}}dx$$$\bf{My\; Try::}$Let $$I = \int\frac{x^4}{(1-x^4)^{\frac{3}{2}}}dx = -\frac{1}{4}\int x\cdot \frac{-4x^3}{(1-x^{4})^{\frac{3}{2}}}dx$$ ... 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)-...
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}{...