integration's questions - Chinese 1answer

42.722 integration questions.

I a trying to solve the following problem: Let $K=\{(x,y,z)\in\mathbb{R}^{3} : |x|+|y|\leq 1,\ z\geq 0,\ x^2 +y^2+z^2\leq 1\}$ Show that $\int_K xz^2 dxdydz = 0$ without using Fubini's theorem. I ...

How to prove the following identity? $$\int\limits_{-1}^1 \frac{(1-u^2)^{\frac{D-4}{2}}du}{1+A u}=\frac{\sqrt{\pi}\Gamma\left(\frac{D-2}{2}\right)}{\Gamma\left(\frac{D-3}{2}\right)} \int\limits_0^1 ...

I know that $ \delta (x) = \frac{1}{2 \pi } \int_{-\infty}^{+\infty} e^{iwx}dw$ and I know indirect methods to solve this. But I can't solve the integral in a direct way. I tried this site to show ...

Approximate integral: $\int_3^4 \frac{x}{\sqrt{(7x-10-x^2)^3}}dx.$ My attempt: Let I = $\int_3^4 \frac{x}{\sqrt{(7x-10-x^2)^3}}dx$ $u=7-x\implies I=\frac 72 \int_3^4 \frac 1{\sqrt{(7x-10-x^2)^3}}dx$...

The integral in question is $$\int(2-x)^2\ln{(4x)}dx$$ I've set $u=\ln{(4x)}$ with $du=\frac{4}{x}dx$ and $dv=(2-x)^2dx$ with $v=-\frac{1}{3}(2-x)^3$ Using integration by parts, I end up with: $$-\...

Let $y(x)$ be the function $$y(x)=\begin{cases}10 &\text{if} \ x=5 \\ 0 &\text{otherwise}\end{cases}.$$ Calculate the integral $$\int_{-\infty}^{\infty} y \ dx,$$ Is the answer 10? Please ...

Calculate $$\iiint_V \frac{1}{(x+y+z)^3} \, dV$$ Where $V$ is the volume bounded by the planes $$\{4x + 3z = 12, 4x + z = 4, 4y + 3z = 12, 4y + z = 4, z = 0\}$$ I've used simple coordinates ...

Integrate $$\int_{-\infty}^{\infty} \cos(\sqrt{2} x)e^{-x^2} dx$$ All i want to know is how to proceed. I have been stuck at this for a very long time now. I tried writing the $\cos$ term as ...

I need a closed-form solution for the following integral: $$\int_{0}^{\infty}\frac{dt}{(t+a)^{m} (t +b)^{n}};\,\,a,b>0;\,\,m,n\geq1\,\text{are integers}$$ If $m$ and $n$ are non-integers, I can ...

My question is somehow related to this question. What I would like to know is why we ignore the fact that integral of derivative of f(x) is equal to ...

I came across a concept which I have never dealt with before. I had an idea but it is flawed and I would like to know why it is flawed. To those who are familiar with that subject it will seem ...

Let $\mathcal{R}([a.b])$ the set of all Riemann-integrable functions in $[a,b]$. Let $\mathcal{R}^{*}([a,b])$ the set of all Generalized Riemann-Integrable functions in $[a,b]$ (I'm talking about the ...

let $1<t<2$. I need to evaluate $$\lim_{\epsilon\rightarrow 0^{+}}\int_{0}^{\infty}\,e^{-\epsilon x}\,\frac{|\sin{x}|^{t}} {x^{t-1}}\,dx$$ If $t>2$ one can easily apply the dominated ...

I know the chain rule for derivatives. The way as I apply it, is to get rid of specific 'bits' of a complex equation in stages, i.e I will derive the $5$th root first in the equation $(2x+3)^5$ and ...

I'm investigating integrals in the form $$I(a):=\int_0^\infty \frac{dx}{e^x+ax}$$ So far, I haven't been able to find any special values other than $I(0)=1$, and I've only managed to evaluate these ...

There is this definition I couldn't quite get my head around. It is the definition of the integral in regard to the simple functions. The integral of the measurable positive function $f:X \rightarrow ...

I begin this post with a plea: please don't be too harsh with this post for being off topic or vague. It's a question about something I find myself doing as a mathematician, and wonder whether others ...

Find the volume of of the wedge shaped solid that lies above the xy plane, below the $z=x$ plane and within the cylinder $x^2+y^2 = 4$. I'm having serious trouble picturing this. I think the z ...

Given the equation below: $u(x) =\frac {1}{2a} \int_{x-a}^{x+a} g(\xi) \, d\xi$ With $$u(x) = \begin{cases} 0,& (|x|>\ 1+a) \\ -\frac 1 a (|x|-a-1),& (1-a\le|x|\le1+a) \\ 2, &(|x|&...

Please correct me if I'm wrong. In terms of Riemann integrability: If we are taking into consideration Riemann integrals on a closed interval, then any continuous function is integrable. In terms of ...

By integration by parts and the substitution $x = \sin t$ we can easily calculate the integral $ \int_{0}^{1} \ln (x+ \sqrt{1-x^2})dx$ which equals to $\sqrt{2} \ln (\sqrt{2} +1) -1.$ I’ve tried to ...

From one of the maths seminar course introduction to calculus for the first year students @King math's school of academic. $$\int_{0}^{\pi/2}\int_{0}^{\pi/2}\frac{\mathrm dx \mathrm dy}{\left(\frac{\...

Consider $$\int_{-\infty}^{+\infty} x \dfrac{\partial}{\partial x}\left(\Psi^*\dfrac{\partial \Psi}{\partial x} - \Psi\dfrac{\partial \Psi^*}{\partial x}\right) dx $$ If I apply integraton by parts ...

I need to find the length of a spiral. The spiral start at a certain radius R1 (25mm) and ends at a larger radius R2(unknown). As the spiral spins outwards, the distance between each arm of the spiral ...

I want to understand the following: Let $\pi:X \to Y$ a fiber bundle and $\omega$ a closed smooth differential form. Define $I: Y\to \mathbb C$, $y\mapsto I(y)=\int_{\pi^{-1}(y)} \omega$. Then Stoke'...

I would like to calculate the following integration: $$\int_{x_1=0}^{2^{R}-1}\cdots\int_{x_N=0}^{2^{R-y_1-\cdots-y_{N-1}}\hspace{0.5cm}-1}\sum_{k=1}^{N} x_k\hspace{0.3cm} dx_N\cdots dx_1,$$ where $$...

Find for how many values of $n$. $I_n=\int_0^1 \frac {1}{(1+x^2)^n} \, dx=\frac 14 + \frac {\pi}8$ My attempt (integration by parts): \begin{align} I_n & = \int_0^1 \frac 1{(1+x^2)^n}\,dx = \...

$$\int\arccos\frac{1-x^2}{1+x^2}\arccos\frac{4-x^2}{4+x^2} \, dx$$

I am working with the following Economic model of labour and consumption decisions: I have a population whose mass is normalized to one of consumers. They derive utility from consumption $c$,...

I am facing a problem which asks to find coordinates of mass center of a homogeneous curve L. Obviously, this should be done by the line integral but the part I do not understand at all is the ...

Calculate $$\int_{T} \vec{A} \cdot \hat{n} \ dS$$ where $\vec{A}=4 \ \hat{i}$, and $T$ is the intersection between $x+y+2z=1$ and $\Gamma=\{(x,y,z): x\ge 0, y\ge0,z\ge 0\}$ My work. Normal to the ...

let $F=\left(\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2}\right)$ and $R(t)=(\cos t,\sin t)$ (the curve is a circle with radius 1) now: \begin{equation} \int_{R}F_1.dx+F_2.dy = \int_{0}^{2\pi}-\sin t\ dt + \...

Find the Value of $$I=\int_{0}^{\infty}\frac{x^3 \: dx}{e^x-1}$$ My try: Put $$e^{-x}=t$$ $$I=\int_{1}^{0} \frac{-(\ln t)^3 \times -dt}{1-t}=\int_{0}^{1} \frac{(\ln t)^3 dt}{t-1}$$ Now using parts ...

In this note Analysis Tools with Examples On page 285, it defines the integration on an embedded hypersurface as follows: The idea is straightforward: Chop a hypersurface into pieces that are ...

suppose that we have a function $F(x,y)=(\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2})$ if we calculate $curlF$ we understand that $curlF=0$ ( computing $curlF$ is not my question). now the book has written ...

Let $$ \rho: \mathcal{X}\times\Theta\rightarrow]0,1] \\ \rho(x, \vartheta) = e^{a(\vartheta)\, T(x)-b(\vartheta)} \, h(x) $$ with: $\Theta\subset\mathbb{R}$ an open interval. $\mathcal{X}$ a Borel-...

Suppose that $f:[0,1]\times [0,1]\rightarrow\mathbb{R}$ has continuous partial derivatives. Define $F:[0,1]\times [0,1]\rightarrow\mathbb{R}$ by $$F(u,v)=\int_{0}^{u} f(v,y) \ dy$$If $u=u(x)$ and $v=...

I got stuck in solving the following indefinite integral. $$\int \sqrt{ 9 \sin^2 t + 4\cos^2 t } \, dt . $$ Please Help. To be honest was definite integral from $0$ to $\pi$. I know it tells the ...

Specifically, for a double integral $$\int_a^b \int_{g_1(x)}^{g_2(x)} f(x,y) \, dy \, dx$$ how would you change the order of integration without having to sketch it out? I came across this while ...

When changing the order of double integrals, I have always relied on sketching the region. I have recently come across this example on MSE by @FelixMartin which seems to avoid visual-based reasoning, ...

Solve $\frac{dy}{dx}=\cos(x+y)+\sin(x+y)$ My Attempt $$ \frac{dy}{dx}=\sqrt{2}\cos(x+y-\tfrac{\pi}{4}) $$ Set $t=x+y-\tfrac{\pi}{4}\implies y=t-x+\tfrac{\pi}{4}$ $$ \frac{dy}{dx}=\frac{dt}{dx}-1=\...

Given a functional $L: X\to \mathbb{R}$. It is possible for two $x,x_h \in X$, to write $$L(x)-L(x_h)=\int_0^1 L'(x_h+s(x-x_h))(x-x_h) ds$$ where $L'(\cdot)(v)$ is the directional in $v$. Why does ...

I want to change the bounds of this integral from $0,2t$ to $0,t$ (for Laplace transform) I try using an extra variable $x=2t$ but I can't realize how to change the $\sin:$ $$ f(t) = \int_0^{2t} \...

I know how to reverse the order of integration and summation and when I can if the integrand is a single sum. But is there a similar rule for reversing the order of summation and integration for a ...

Consider the following integration: $$ I(x_0|c, d) = \int_{x=-\infty}^{x_0} (cx - d) \mathcal{N}(x; \mu, \sigma^2) \,dx $$ In this notation, $\mathcal{N}(x; \mu, \sigma^2)$ is a normal distribution: ...

Consider Tommy’s integrals: $$a) \int_1^{\infty} \frac{\operatorname{li}(x)^2 (x - 1)}{x^4}\, dx = \frac{5 }{36}\pi ^2 $$ $$ b) \int_1^{\infty} \frac{\operatorname{li}(x)^2 (x - 1)}{x^5}\, dx = \...

So i have this integral tending from 2(on top) to 1(on bottom ) of $$\int_1^2x\ln(x^2 +3)x\,dx$$ If I turn it into $x^2\ln(x^2 +3)\,dx$, with integration by parts I get $$\frac{16}{3}\,\ln(2) + \...

The question is to solve the integral $$\iint_D e^\frac{x+y}{x-y}dxdy$$ with the substitutions u=x+y and v=x-y, where D is the trapezoidal region with vertices in (1, 0), (2, 0), (0, −2) and (0, −1). ...

I'm looking for a book that treats multiple Riemann integration directly in $\mathbb{R}^n$ in particular the demonstration of change of variables. Thanks, sorry for bad english

How do we evaluate $\int\frac{1}{\ln{x}}dx$? Is it a special integral, or some simple function, or what substitution will help?

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