Tagged Questions

All aspects of integration, including the definition of the integral and computing different types of integrals. For questions solely about the properties of integrals, don't use this tag alone! Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another ...

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0
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0answers
6 views

Derivative of integral of sign function

Is there any vlaues for $a(\int x,x,\dot x)$, $b(\int x,x,\dot x)$ and $c(\int y,y,\dot y)$ such that the following relation holds $\frac{d}{dt}\int_{a(\int x,x,\dot x)}^{b(\int x,x,\dot ...
5
votes
0answers
41 views

An integral by O. Furdui $\int_0^1 \log^2(\sqrt{1+x}-\sqrt{1-x}) \ dx$

The following integral was proposed in a paper by O. Furdui, namely $$\int_0^1 \log^2(\sqrt{1+x}-\sqrt{1-x}) \ dx$$ and then the generalization $$\int_0^1 \log^2(\sqrt[k]{1+x}-\sqrt[k]{1-x}) \ ...
0
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0answers
15 views

Integral of $\int_{0}^{z^*}0dz$

So I have to find the max upward velocity for an air parcel. Anyways so $U$ is the vertical velocity and $z^*$ is the height were the max vertical velocity occurs. From this I got the integral of ...
1
vote
2answers
40 views

Am I doing something wrong with this improper integral?

I have a little discussion with my friends about my "resolution" and calculation of $$\int_{-\infty}^1 e^{4x} \, dx.$$ I did $$\int_{-\infty}^1 e^{4x} \, dx =\int_{-1}^{\infty} e^{-4x} \, dx = ...
7
votes
2answers
51 views

How to prove $\int_{0}^{-1} \frac{\operatorname{Li}_2(x)}{(1-x)^2} dx=\frac{\pi^2}{24}-\frac{\ln^2(2)}{2} $

$\def\Li{\operatorname{Li}}$ I wonder how to prove: $$ \int_{0}^{-1} \frac{\Li_2(x)}{(1-x)^2} dx=\frac{\pi^2}{24}-\frac{\ln^2(2)}{2} $$ I'm not used to polylogarithm, so I don't know how to tackle it. ...
1
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0answers
39 views

Computing an integral explicitly

I have the following integral: let $k$ be a positive fixed integer and $\varepsilon \in (0,1/k)$. Let $t>0$ and $r>0$. We consider $$\int_0^1 \int_0^{t/2} (t^{2\varepsilon} - ...
1
vote
1answer
39 views

How to approximate this nasty exponential function with an integral?

What is the best way to approximate a function of the following form, $$ \text{exp}\left(-\int_{y}^{+\infty} f(x)\ dx \right)$$ Any approximation to this, does taylor series work? The reason I am ...
1
vote
3answers
79 views

Integrate $ \int_0^{\infty} \! x^2 e^{-ax^2} \, \mathrm{d}x $ [on hold]

Integrate $$ \int_0^{\infty} \! x^2 e^{-ax^2} \, \mathrm{d}x $$ We may assume without proof: $$ \int_0^{\infty} \! e^{-x^2} \, \mathrm{d}x = \frac{\sqrt{\pi}}{2}$$
1
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0answers
14 views

The inverse of the integration over a ball with radius $\epsilon$

First of all sorry for the nondescript title, but this seemed like the most suitable one. Now let $d\geq2, D\subset \mathbb{R}$ a domain and $G:D\times D\rightarrow[0,\infty]$ continuous. Define ...
1
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3answers
272 views

integral calculation is wrong. Why?

$$\int \sqrt{1-x^2} dx = \int \sqrt{1-\sin^2t} \cdot dt= \int \sqrt {\cos^2 t} \cdot dt= \int \cos t \cdot dt = \sin t +C = x +C$$ The answer is wrong. Why?
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1answer
56 views

Changing the limits of integration when function is infinite

Suppose I have an integral: $$\int_0^{\pi/2}f(b\tan x)dx$$ With $b$ being some positive parameter. Now if I want to change variables in this way: $b\tan x = \tan t$, how will the upper limit change? ...
0
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0answers
17 views

Integrating over a power of the infinitesimal

I don't know if the title makes sense (or if the question makes sense at all for that matter) but here I go. Suppose I have a piecewise constant function $y=f(x)$ with $x,y\in\mathbb{R}^+$, described ...
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0answers
11 views

Plot Cartesian prism in cylindrical system

My objective: Use cylindrical coordinates to find the volume of the prism whose base is the triangle in the xy-plane given by y = 0, x = 1, and y = $\frac{7}{2} x$, and whose top is given by z = ...
1
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0answers
23 views

Expand $\int_{-1}^0 e^{a\cos{\theta}}J_0(b\sin{\theta})\,d\cos{\theta}$ in spherical harmonics.

I want to solve the integral (a probability density function) $$ g(\gamma)=\int_{-1}^0 e^{-f\cos{\theta}\cos{\gamma}}J_0(-if\sin{\theta}\sin{\gamma})\,d\cos{\theta} $$ numerically, everything is ...
2
votes
4answers
78 views

Evaluate $\int_0^2\frac{x^5}{\sqrt{x^3+6}}\,dx.$

I am stuck on the following integral: $\displaystyle\int_0^2\dfrac{x^5}{\sqrt{x^3+6}}\,dx.$ I have no idea how one can work it out. Normally I'd try $u=x^3+6$ but this surely does not work here.
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1answer
29 views

Finding a Counter Example - Limits of integrals of an increasing sequence of Borel measurable functions

I need to find a counter example to the following problem. I'm trying to think of some, but maybe I'm not creative. I'm not sure. Let $h$ and $h_1, h_2, h_3, ...$ be Borel measurable functions such ...
0
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1answer
29 views

integration problem involving substitution [on hold]

$$\int \frac{x^2}{\sqrt{x^3 -1}}\, dx$$ How can i solve this problem if i want to let $u= \sqrt{x^3 -1}$ Thank you
7
votes
2answers
92 views

Finding the closed form of $\int_0^1 \frac{(1-x+x\log x)\, \ Li_3(x)}{x(x-1) \log x} \ dx$

$\def\Li{{\rm{Li}}}$Here I have a question I just received, and still trying to find a proper starting point $$\int_0^1 \frac{(1-x+x\log x)\Li_3(x)}{x(x-1) \log x} \ dx$$ What starting point would ...
2
votes
1answer
65 views

how to integrate $\mathrm{arcsin}\left(x^{15}\right)$?

Integral by parts: $$ I = x\sin^{-1}\left(x^{15}\right) - \int\frac{15x^{15}}{\sqrt{1-x^{30}}}dx $$ then what? The answer by wolfram gives an answer contains hypergeometric ${}_2F_1$ function,because ...
1
vote
1answer
20 views

Prove the f is integrable when $f(x)=(1-x^4)^\frac{1}{2}$

Let $f(x)= (1-x^4)^{1/2}$ by $f:[0,1]\to \mathbb{R}$. I need to prove the $f$ is integrable on $[0,1]$. I think I need to use a partition but I have no idea how to prove the integrability or where to ...
0
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0answers
17 views

A question related to “supremum”

Let $f\in L_{p}([0,1])$ and $[a,c]\subset [0,1].$ For any $b\in (a,c)$ is the following equality true? $$ \underset{|h|\leq b-a}{\sup}\int_{a}^{b}|f(x+h)-f(x)|^{p}dx+ \underset{|h|\leq ...
10
votes
2answers
92 views

Evaluation of a tough double integral

This is an integral coming from personal research, and very important to me, but it does not seem an easy job to do. If a solution is not possible then I'd be glad with a closed form only. ...
0
votes
1answer
28 views

Am I correct with this change of variable?

I have been solving a problem from a paper I read related to poisson point processes and for some reason I am not reaching the same result the paper has. The problem is re-expressing an expression by ...
0
votes
1answer
21 views

Find the volume of the solid generated by revolving the region bounded by $y=x$ and $y=x^2$ about the line $y=x$

Find the volume of the solid generated by revolving the region bounded by $y=x$ and $y=x^2$ about the line $y=x$ I am confused, how do we approach such problems, where the rotation lines are not ...
0
votes
1answer
22 views

Splitting Up Integrals and Multiplying Them

$$I_x = \int_0^b\int_0^h\rho y^2\,\mathrm{d}y\mathrm{d}x$$ So here's the current problem I'm working on, just for an example. I saw my teacher break up a triple integral in class today then multiply ...
0
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1answer
20 views

Volume of a solid sphere hole

A round hole of radius $\sqrt{3}$ is bored through the centre of a solid sphere of radius 2. Find the volume of the material removed . Looking for a clever way to solve this problem
0
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0answers
88 views

Nasty integration?

So I am trying to solve the following integral and apparently its not integrable or I might be wrong. Not even computer software can integrate. Can anyone tell me if this is integrable or not? The ...
1
vote
0answers
12 views

Integrate Two Dot-Product-Power Terms

I need to compute something like the following integral:$$ \int_{\Omega_\vec{a}} \left< \vec{x} \cdot \vec{b} \right>^n \left(\vec{x}\cdot\vec{a}\right)^m d\vec{x} $$Notational issues: The ...
0
votes
1answer
22 views

Radon transforms and determining a separable function

I am interested in the Radon transformation for separable functions $F(x,y) = f(x)g(y)$. Why is it in tomography that a separable function is determined completely by two of its projections ? And ...
0
votes
0answers
11 views

Integrate determinant of product of two matrices

Let $V\left(i,j\right) = \alpha_j^{i-1}$ be the $\left(i,j\right)^{th}$ element of the matrix $V\in\mathbb{R}^{n\times n}$. Such matrices are called Vandermonde matrices. Let $X = \left|V\times ...
0
votes
0answers
13 views

Saddle point method for asymptotic expansion

Any idea on how to derive an asymptotic expansion of the following integral for $x$ around zero (using Saddle point method): $$\int_{0}^{x}\left(u+\alpha\right)^{-a - \left(b-a\right)e^{-\lambda ...
1
vote
2answers
50 views

Why is $F(x)$ continuous at $x=0$?

Let $$f(x) = \left\{ \begin{array}{ll} x & \mbox{if } x < 0 \\ \sin x & \mbox{if } x \ge 0 \end{array} \right.$$ $F(x)$ the anti-derivative should be $\frac{x^2}{2} + C_1$ for ...
1
vote
4answers
51 views

Derivative of $\int_{x}^{0} \frac{\cos xt}{t} dt$

I am working on the following problem: Find the derivative of $f(x)=\displaystyle \int_{x}^{0}\displaystyle\frac{\cos xt}{t}dt$. The answer I am supposed to get is $\displaystyle ...
0
votes
1answer
19 views

Volume of region inside a surface

Find the volume of the region inside the surface $z = x^2 + y^2$ and between $z = 0$ and $z = 10$. Really the only thing I need help with in this problem is setting up the limits of integration. ...
0
votes
0answers
22 views

Finding an integral for a given Riemann Sum

Take the Riemann sum: $= \displaystyle \lim_{m\to\infty} \frac{1}{m}\sum_{x=1}^{m} me^{-x}$ How can someone convert that into an integral? We know $\Delta(x) = \frac{1}{m}$. So $me^{-x}$, is the ...
0
votes
1answer
23 views

Nonconvergent convolution integral

I have a convolution integral where $F(\tau) = F_0$ and $g(t - \tau) = \sin(\omega_n(t-\tau))$ so $$ F_0\int_{t_0}^{\infty}\sin(\omega_n(t-\tau))d\tau $$ which doesn't converge. Can I do the ...
0
votes
0answers
15 views

Double Integral Mistake with Parametric Equation

I'm trying to figure out the mass of an object bounded by $y=0$ and $y=\sqrt{1-x^2}$ the density at a given point is proportional to its distance from the origin; $\rho(x,y) = kxy$. So I set it up ...
2
votes
0answers
71 views

Is there a function whose definite integrals are all 0?

Is there a continuous function $f: [0,1] \rightarrow \mathbb{R}$ such that $f(x) \neq 0$ for some $x \in [0,1]$ and, if we define $F_n(x) = \int_{0} ^ {x} F_{n-1}(t) dt $ (where $F_0(x)=f(x)$), then ...
0
votes
1answer
26 views

Numerically solve integral with a function as variable of integration

I want to use a function as variable of integration for example in evaluating the integral: $\int_0^1 e^{\cos x}f(\sin x)d\cos x$ in which $f(x)$ is an arbitrary function.
0
votes
0answers
16 views

Integeration of 2nd derivative

I have a question regarding solving the integral of the partial. Here is the equation: $$R_m = \int \psi^m \frac{\partial^2}{\partial\psi^2} \left[\left\langle \epsilon_\phi | \psi \right\rangle ...
2
votes
1answer
48 views

Why am I obtaining an imaginary part for my integration

I try to solve an integration as follows, $$\int \frac{sy^{-1}}{(1+sy^{-1})} \text{exp}(-\sqrt{y})dy$$ as you can see its pretty complicated, and I get an answer like the following using Wolfram ...
1
vote
1answer
36 views

Spectral Measures: Domain Criterion

Given a topological space $\Omega$ and a Hilbert space $\mathcal{H}$. Let $\mathcal{B}(\Omega)$ be its Borel algebra and $\mathcal{B}(\mathcal{H})$ its bounded operators. Moreover, given a spectral ...
2
votes
2answers
20 views

Why is $\vec{s}=\frac{\vec{r}}{V^\frac{1}{3}} \Leftrightarrow d\vec{s}=\frac{d\vec{r}}{V}$?

I am following a course which contains a part in statistical thermodynamics. One of the questions involves the partition function $Q_N$. I could not figure out the answer of the question myself, so I ...
12
votes
1answer
120 views

Evaluating by real methods $\int_0^{\pi/2} \frac{x^5}{2-\cos^2(x)}\ dx$

I'm sure you guys can briefly get the result by some methods of complex analysis, but now I'm only interested in real analysis methods of proving the result. What would you propose for that? ...
0
votes
0answers
11 views

Integration of the reciprocal of sum exponential

Any one know the method to do the integration as $$\int\frac{x^2\cdot \exp (-ax^2) \exp(-bx^2)}{\exp(-ax^2)+\exp(-bx^2)}dx$$ It can be simplified as $$\int\frac{x^2}{\exp(ax^2)+\exp(bx^2)}dx$$ ...
0
votes
0answers
18 views

Asymptotic expansion at infinity of integral function

Given $q\in(0,1)$ find $z$ such that $$ F(z)\equiv\int_{-\infty}^{z}\frac{e^{-\frac{y^2}{2 \sigma _{22}^2}} \text{erfc}\left(\frac{\rho \sigma _{11} y-\sigma _{22} V}{\sqrt{2-2 \rho ^2} \sigma ...
1
vote
2answers
36 views

meaning of integration

I read that integration is the opposite of differentiation AND at the same time is a summation process to find the area under a curve. But I can't understand how these things combine together and ...
-1
votes
0answers
20 views

surface and cone integrals [on hold]

can someone take me through these two questions, I have the answers but not the steps and I have no idea how to even get started, thanks!
3
votes
1answer
39 views

Integrating $\int \sec^2(x) \tan(x) dx$ by trig substitution

I know I am supposed to integrate $$\int \sec^2(x) \tan(x) dx$$ by substituting $u = \tan(x)$ and get $du = \sec^2(x)$. However, why can't I use $u = \sec(x)$, $du = \tan(x) \sec(x)$?
0
votes
1answer
17 views

Poisson integral with discontinuous $U$

Let $U$ be a piecewise continuous function and bounded for all real numbers. Then define the Poisson Integral for the UHP to be (It can be deduce from the one for the unit circle). ...