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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10
votes
2answers
282 views

Is this sequence bounded ? (An open problem between my schoolmates !)

Let $f$ be a smooth function (say $\mathcal{C}^{\infty}$) in its two real variables ($t$ and $T$). I consider the following sequence defined by $$A_n:=\lim_{T \to \infty} \int_{0}^{1} e^{-n t} ...
2
votes
0answers
75 views

Shannon Entropy, prove $H(Wx)=H(x)+\log|\det W|$

I'm doing an essay on ICA (independent component analysis), and I could use some help. In essence, ICA is an algorithm that minimizes the entropy of $n$ $1$-dimensional random variables, but to show ...
1
vote
1answer
39 views

If an absolutely continuous function satisfies $f(0)=0$, then $\int_0^x|f(t)f'(t)|dt\leq\frac{1}{2}\big(\int_0^x|f'(t)|dt\big)^2$

Assume that $f$ is absolutely continuous on $[0,a]$ for every $a\geq0$, and $f(0)=0$. Prove that \begin{equation*} \int_0^x|f(t)f'(t)|dt\leq\frac{1}{2}\biggl(\int_0^x|f'(t)|dt\biggr)^2 ...
3
votes
2answers
108 views

Finding $\int \frac{1+\sqrt{\frac{x+1}{x-1}}}{1-\left(\frac{x+1}{x-1}\right)^{\frac14}}dx$.

I am trying to integrate $$\int \frac{1+\sqrt{\frac{x+1}{x-1}}}{1-\left(\frac{x+1}{x-1}\right)^{\frac14}}dx$$ I tried substitutions such as $y=\left(\frac{x+1}{x-1}\right)^{\frac14}$. But it doesn't ...
1
vote
1answer
28 views

Integral with possibly partial fraction

I am trying to solve $\int \frac{x+1}{x^2+3}dx$. How would I do it? If I had $x^2+3$ in denominator I would proceed to look for A and B in $\int \frac{x+1}{x^2-3}dx=\int ...
4
votes
2answers
101 views

Calculating Harmonic Sums with residues.

Evaluate: $$\sum_{n=1}^{\infty} \frac{H_n}{(n+1)^2}$$ A user stated: "most of the time sum up the residues of $(\gamma+\psi(z))^2\cdot r(z)$. To determine the residues, just expand the digamma ...
2
votes
0answers
85 views

Evaluating a sum $-\zeta'(2)$

Is it possible to obtain any closed-form expression for the infinite sum $$\sum_{n=1}^{\infty}\frac{\log(n)}{n^{2}}$$ by Residue calculus? My thought was to try to integrate $$f(z) ...
0
votes
0answers
29 views

Integration by parts and gamma functions

I am working through something to deepen my understanding and I am not sure of this step. Getting from here $$=\int^\infty_y \frac{e^{-z} z^{n+v-1}}{\Gamma(n+\nu)} \int^\infty_y \frac{e^{-k} ...
0
votes
0answers
31 views

Residues of a digamma based function.

I was wondering how we can find the residues of the digamma function, From: Integral Calculation Find the residues of: $$f(z) = \frac{(\gamma + \psi(-z))^2}{(z+1)(z+2)^3}$$ The answer is in the ...
3
votes
1answer
85 views

Integration with Beta Function $\beta$ [closed]

Given that: $$\int_{c\ -\ j\infty}^{c\ +\ j\infty}\left({\sigma\,x^{-1}}\right)^u\beta\left(u,a\right)du=\left(1-{x \over \sigma}\right)^{a-1}$$ whereby $\sigma>0$, $a>0$ and $x$ is a real ...
3
votes
6answers
201 views

Getting wrong answer trying to evaluate $\int \frac {\sin(2x)dx}{(1+\cos(2x))^2}$

I'm trying to evaluate $$\int \frac {\sin(2x)dx}{(1+\cos(2x))^2} = I$$ Here's what I've got: $$t=1+\cos(2x)$$ $$dt=-2\sin(2x)dx \implies dx = \frac {dt}{-2}$$ $$I = \int \frac 1{t^2} \cdot \frac ...
1
vote
1answer
230 views

Integrating a cross product.

I am given that $\mathbf{k} . m \mathbf{q} \wedge \mathbf{\ddot{q}} = 0$ and my book says that integrating this wrt time gives $\mathbf{k}.m \mathbf{q} \wedge \mathbf{\dot{q}} = $constant. I don't ...
0
votes
0answers
25 views

Proving an equality related to Lebesgue integration

I am proving some facts in probability but stuck in showing the following fact: Notation: For any set $B \subseteq \mathbb{R}$, $y \in \mathbb{R}$, let $y-B := \{y-b : b \in B \}$. Also, for any set ...
0
votes
1answer
90 views

Evaluate: $\int e^{x^4}(x+x^3+2x^5)e^{x^2} dx$ [closed]

Evaluate: $$\int e^{x^4}(x+x^3+2x^5)e^{x^2} dx$$ I know the answer of this integral but got stuck at how to solve this. It seems to be the form like $ \int e^x(f(x)+f''(x))dx = e^x f(x)+C$
0
votes
0answers
45 views

Show that $\int_b d\omega=0$ where $b(s,t)=\Phi_s(c(t)) $

Let $c:[0,1]^k \rightarrow \mathbb{R}^n$; $t \mapsto c(t)$ be k-cell with $k < n$. Let $\mathbb{Y}$ denote a vector field on $\mathbb{R}^n$ with flow $\Psi_s$. Define a $(k+1)$-cell $b:[0,1]^{k+1} ...
1
vote
1answer
111 views

Finding an outward pointing normal on the unit sphere,

I am trying to apply the divergence theorem, but I need to find an outward pointing normal vector on the unit sphere. The answer gives $\hat n= (x_1,x_2,x_3)$. Is the person who wrote up the ...
2
votes
1answer
43 views

General Solution to Almost Riccati Like Equation

Consider the differential equation $$ y' = a_0(x) + a_1(x)y + a_2(x)\frac{1}{y}$$ I am attempting to find the general solution to this. One thing I can note is that the entire equation can be ...
7
votes
2answers
171 views

Asymptotics of $\int_{0}^{+\infty}\!\!\frac{dx}{\sinh^2(\epsilon \sqrt{x^2+1}) } $ for $\epsilon$ near $0$

How to find an asymptotic expansion, for $\epsilon$ near $0$, of the following integral $$ I(\epsilon):=\int_{0}^{+\infty}\frac 1{\sinh^2 (\epsilon \sqrt{x^2+1}) } {\rm d}x. $$ As $\epsilon ...
0
votes
1answer
108 views

tetrahedron volume in the first octant

The surface is given: xyz = 2 It is in the first octant so x > 0, y > 0, z > 0. The tangent plane taken at any point of this surface binds with the coordinate axes to form a tetrahedron. Task: ...
3
votes
2answers
85 views

Faulty application of the Fundamental Theorem of Calculus to $f(x) = 0$ for $x\ne 0$, $f(0)=1$

I think I have given a fallacious proof but I can't seem to find what is wrong with it. Suppose $f : \mathbb{R} \rightarrow \mathbb{R}$ has the property that $\forall a,b \in \mathbb{R}. \int_a^b ...
1
vote
1answer
38 views

complex analysis fundental theorem of caculus

Can anyone please explain how $$\int \frac{1}{(z-2)^3}dz $$ evaluated about the closed continuous path $$1+3e^{i2t\pi}$$ is 0 by the fundamental theorem of calculus?
0
votes
1answer
48 views

Uniform continuity of the antiderivative

We know that if $f:\mathbb{R}\to\mathbb{R}$ is a function such that $$\sup_{x\in\mathbb{R}}|f(y)|<\infty,$$ then the function $g(x)=\int_0^xf(y)dy$ is uniformly continuous. I am just wondering ...
1
vote
0answers
34 views

Proving that $\frac{d}{dt}\int \Phi_t^*\omega=\int_{\Phi_t \circ \partial c} i_{\mathbb{X}}\omega$

Let $\omega$ be a closed $k$-form on $\mathbb{R}^n$ and $c:I^k \rightarrow \mathbb{R}^n$ a $k$-cube on $\mathbb{R}^n$. Let $\mathbb{X}$ be a vector field on $\mathbb{R}^n$ with flow $\Phi_t$. Show ...
1
vote
1answer
86 views

Evaluate Integral $\int_{c\ -\ j\infty}^{c\ +\ j\infty} \left({\sigma\,x^{-1}}\right)^s{\Gamma(\beta_1-1+s)\over \Gamma(\beta_1+\beta_2-1+s)}\,ds$

I am at this point of integration where: $$\int_{c\ -\ j\infty}^{c\ +\ j\infty} \left({\sigma\,x^{-1}}\right)^s{\Gamma(\beta_1-1+s)\over \Gamma(\beta_1+\beta_2-1+s)}\,ds$$ whereby $\beta_1$, ...
0
votes
1answer
49 views

Find the volume of the solid bounded by $ y=0 , z=1-x^2, z=x^2-1, y+z=1 $

i need to know the first steps and strategies of solving this kind of problems i know that i need to find the limits of x and y and then do the integral of a certain functions by dxdy
4
votes
2answers
35 views

how to integrate with square root with mode?

I want to integrate $$\int_{-1}^{2}\sqrt{|x|}dx$$ But I dont know how to do it should I integrate it fist from $-1$ to $0$ and then from $0$ to $2$ making one of equation in minus other in plus?
2
votes
1answer
35 views

Prove that $\int_{X}\exp(f(x))d\mu(x)\cdot\int_{X}\exp(-f(x))d\mu(x)\geq\mu(X)^{2} $

Let $(X,\mathfrak{S},\mu)$ a measure space, and $f:X\rightarrow \mathbb{R}$ a measurable function. I have to prove that $$\int_{X}\exp(f(x))d\mu(x)\cdot\int_{X}\exp(-f(x))d\mu(x)\geq\mu(X)^{2}.$$ I ...
6
votes
1answer
96 views

How prove $ \int_0^1 f(x)dx-\exp\left(\int_0^1\log(f(x)) dx\right)\le\max_{0\le x,y\le 1}\left(\sqrt{f(x)}-\sqrt{f(y)}\right)^2 $

Consider a continuous function $f:[0,1]\to\mathbb{R}^{+}$. How show that $\int_0^1 f(x)dx-\exp\left(\int_0^1 \log(f(x)) dx\right)\le \max_{0\le x,y\le 1}\left(\sqrt{f(x)}-\sqrt{f(y)}\right)^2$?
0
votes
1answer
27 views

prove that $∀ε>0∃p∈P(U(f,p)−L(f,p)<ε)$

$F:[0,1]\times[0,1]\longrightarrow R$ $ f(x)= \begin{cases} 1, & \text{y<x} \\ 0, & \text{y $\geqslant$x} \end{cases} $ i have a problem choosing my p∈P and proving the statement any ...
1
vote
1answer
101 views

Cauchy's Residue Theorem for Integral $\int_{c\ -\ j\infty}^{c\ +\ j\infty} \left({\sigma \over x}\right)^s{{1-\beta^{s+1}}\over s(s+1)}\,ds$

This is a similar problem to the one I posted here. I am at this point of integration where: $$\int_{c\ -\ j\infty}^{c\ +\ j\infty} \left({\sigma \over x}\right)^s{{1-\beta^{s+1}}\over s(s+1)}\,ds$$ ...
2
votes
2answers
80 views

How can I deduce the value of $\frac{1}{\sqrt{4\pi t}}\int_{-\infty}^{\infty}\sin(y)e^{-\frac{(x-y)^2}{4t} } dy$ without actually evaluating it?

How can I deduce that $$ \frac{1}{\sqrt{4\pi t}}\int_{-\infty}^{\infty}\sin(y)\,e^{-\frac{(x-y)^2}{4t} } dy = e^{-t} \sin(x) $$ without actually evaluating the definite integral?
4
votes
1answer
209 views

Evaluate Complex Integral with $\frac{\Gamma(\frac{s}{2})} {\Gamma\big({\beta +1\over 2} - {s\over 2}\big)}$

I am proving this integral: $$ \int_{c\ -\ j\infty}^{c\ +\ j\infty} \left(\,x^{-1}\sigma\beta^{1 \over 2}\,\right)^{s}\ \Gamma\left(\,s \over 2\,\right) \Gamma\left(\,{\beta +1 \over 2} - {s \over ...
3
votes
1answer
90 views

Prove that $\int_0^{\infty} \int_0^{\infty} e^{-(x^2+y^2+2xy \cos \theta)} \,dx dy = \frac{\theta}{2\sin\theta}$

Prove that the following integral: $$\int_0^{\infty} \int_0^{\infty} e^{-(x^2+y^2+2xy \cos \theta)} \,dx dy = \frac{\theta}{2\sin\theta}$$ The hints written on the book are beta function and to ...
3
votes
1answer
93 views

What is the value of this integral (using the Argument Principle),

F(z) is given by $$F(z) = e^zz^{-2}(z-1)(z^2-4)(z+8)^7$$ What is the value of the integral $$\int_0^{2\pi} \frac{F'(3e^{i\theta})}{F(3e^{i\theta})}d\theta \space \space ?$$ I think the relevant ...
2
votes
1answer
82 views

Is this a convolution?

I have the following integral \begin{align*} \int_{-\infty}^\infty f(t) q(t+ax) dt \end{align*} where a is some constant. This integral look a lot like convolution (or correlation). My question is ...
2
votes
1answer
101 views

Cauchy's Residue Theorem with Multiple Gamma Functions

I previously posted a similar problem here and here. This time however I am dealing with multiple gamma functions. This is the problem I am dealing with right now: $$ \int_{c\ -\ j\infty}^{c\ +\ ...
4
votes
2answers
111 views

Methods of evaluating $\int_0^{\infty}\frac{{\rm d}x}{x^2+1}$

Methods of evaluating $$\int_0^{\infty}\frac{{\rm d}x}{x^2+1}$$ Firstly i know that directly: $$\int_0^{\infty}\frac{{\rm d}x}{x^2+1}=\arctan x\Bigg|_{0}^{\infty}=\frac{\pi}2$$ Also we can use the ...
3
votes
2answers
56 views

How to use Stokes Theorem to evaluate $\int_{S} \text{curl} F\cdot d\mathbf{S}$

Let F = $( yz, 0, x)$ and $S$ is the portion of the plane ${x\over2} + {y\over3} + z = 1$ where $x, y, z \ge 0$, oriented with an upward pointing normal then prove: $$\int_{S} \text{curl} F\cdot ...
3
votes
3answers
169 views

General Solution for the Gravity Between Two 3D Triangles

I would like to find the general solution for the gravity between two (flat) triangles in 3D, including the location $(x,y,z)$ where this force should be applied (in order to later account for ...
2
votes
3answers
176 views

It takes 80J of work to stretch a spring 0.5m from its equilibrium position. How much work is needed to stretch it an additional .5m?

It takes $80\,\textrm{J}$ of work to stretch a spring $0.5\,\textrm{m}$ from its equilibrium position. How much work is needed to stretch it an additional $0.5\,\textrm{m}$? Here's what I have: ...
2
votes
1answer
709 views

Moment of inertia of a solid cone about a diameter of the base as axis

I have a solid cone base radius r, height h, mass M and I need to find the moment of inertia about a diameter of the base as axis. The book's answer is $\frac{1}{10}Mh^2+\frac{3}{20}Mr^2$ however, I ...
-1
votes
1answer
81 views

How to find the value of $\int_0^1 \frac{x-1}{\log x}\,dx$? [duplicate]

I want to find the value of $$\int_0^1 \frac{x-1}{\log x}\,dx$$ where $\log x$ stands for the natrual logarithm. I put it in the wolfram alpha, and it saids it's $\log 2$. (Refer to : ...
0
votes
1answer
42 views

$\int_{(0,\infty)}\frac{1}{x}\sin (\frac{\pi}{2}x) d\mu$ for certain measure $\mu$.

I have to check if $f(x):=\frac{1}{x}\sin (\frac{\pi}{2}x)$ is integrable in $(0,\infty)$ , with the measure $\mu$ where $\mu(A)=\mathrm{card}(A\cap\mathbb{N})$ if $A\cap\mathbb{N}$ is finite and ...
8
votes
6answers
308 views

Integration by differentiating under the integral sign $I = \int_0^1 \frac{\arctan x}{x+1} dx$

$$I = \int_0^1 \frac{\arctan x}{x+1} dx$$ I spend a lot of my time trying to solve this integral by differentiating under the integral sign, but I couldn't get something useful. I already tried: ...
0
votes
2answers
38 views

Integrating the absolute of the cosine

For some reason, I do not understand when computing the the integral of |cos(x)| from -pi to -pi/2 gives 1. When i compute it i get -1. There must be something I haven't understood.
1
vote
0answers
61 views

How to evaluate $\int_{0}^\infty \frac{{x^2}}{e^{\beta {\big(\sqrt{x^2 + m^2}}- \nu\big)} + 1} dx$

$$I = \int_{0}^\infty \frac{{x^2}}{e^{\beta {\big(\sqrt{x^2 + m^2}}- \nu\big)} + 1} dx = ?$$ If a constant is added to the exponential in the denominator along with a square root in the exponent and ...
5
votes
3answers
242 views

Integration $\int_{m}^\infty {x}{\sqrt{x^2 - m^2}}e^{-\beta x} dx$

How can the following integration be performed? Does it involve Bessel functions?$$\int_{m}^\infty {x}{\sqrt{x^2 - m^2}}e^{-\beta x} dx$$ EDIT: Actually, the original question is: $$\int_{0}^\infty ...
1
vote
1answer
86 views

An integral with the $\Gamma$ function: $\int_{c- i\infty}^{c+i\infty} u^{s}\:\Gamma(\beta +s-1) \:ds$

I previously posted a similar problem here and I have solved many of the problems from the answers given with explanations. This time however I am at this point of integration where: $$\int_{c\ -\ ...
0
votes
1answer
181 views

Expectation of a discrete random variable: how to convert an integral to a sum?

According to wikipedia and all my textbooks, we define the expectation of a random variable on a probability space $(\Omega, \mathcal{F},P)$ : \begin{align} E(X) &= \int_{\Omega}XdP\\ \end{align} ...
0
votes
3answers
26 views

Integration substitution: How do you find the derivative of the denominator?

I have to integrate: $$ \int_1^2 \frac{37 x}{x^2-6 x+10} \, dx $$ $$ =37 \int_1^2 \frac{x}{x^2-6 x+10} \, dx $$ Then Wolfram Alpha tells me to rewrite the integrand as $$ \frac{2 x-6}{2 ...