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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22 views

A sufficient condition on a real smooth function

Let $f : [0, \infty) \to \mathbb{R}$ be a smooth function. I would like to find a sufficient condition on $f$ in order to have that $$ \liminf_{t\rightarrow \infty} \int_0^t \Big(\frac{t - s}{s} \Big)...
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1answer
20 views

Rotaion Surfaces and Complex Numbers

Consider a continuous invertible map $\varphi:\mathbb{R}^+ \longrightarrow \mathbb{R}$, and define the follwing surface $$ s:\mathbb{C} \longrightarrow \mathbb{R} \times \mathbb{C} $$ $$ \qquad xe^{i\...
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3answers
63 views

Why is dividing by $dx$ or some other differentitator in an integral considered taboo?

Forgive my ignorance but I remember when I was learning calculus, I remember that when we integrate, we always multiply the differentiator to $F(x)$. However, it was never explained to me why we ...
0
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1answer
45 views

integration of exp(cos(x-a)) dx

I would like to compute either of the following integrals: $$\int e^{\cos(x-a)} \, dx$$ or $$\int_{-\pi}^{\pi} e^{\cos(x-a)} \, dx$$ In both cases, $a$ is a constant. MATLAB doesn't seem to be ...
1
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1answer
32 views

Evaluate the integral $\int_c^{\infty}(1-e^{-abc^{m}x^{-m}d})xdx$

I need to know how to solve the following integration $$\int_c^{\infty}(1-e^{-abc^{m}x^{-m}d})xdx$$ where $a,b,c,d,m$ are all greater than $0$. I will be very grateful for your help.
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0answers
31 views

Is it possible to find an approximate or an exact answer for this integral?

I am trying to solve this integral $$ \int\limits_{x_{0}}^{\infty}\frac{2}{x \sqrt{\frac{x^2 }{x_{0}^2}\Big(1-\frac{1}{x_{0}}\Big)-\Big(1-\frac{1}{x}\Big)+\epsilon \left((x-1)^{\nu }-\frac{x^2 }{...
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0answers
32 views

An integral with the Gamma function

Let $c$ be a positive constant. Can you evaluate the following integral? $$ \int_0^{\infty} \frac{x^{cx+1}}{\Gamma(1+cx)}\mathrm{e}^{-x}\,\mathrm{d}x $$ Ps. Do you have some advices or tricks to ...
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0answers
50 views

area of a rectangle

I know that to use polar coordinates instead of $dxdy$ we have $dA=rdrd\theta$. As such, we can have a double integral like $$ \int_{\theta=a}^{\theta=b}\int_{r=c(\theta)}^{r=d(\theta)}f(r\cos\theta,r\...
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0answers
15 views

Any simplifications possible for the following integral?

Here is an integral $$\int_0^a\mathop{\mathrm{d}z}f(z)\mathrm{e}^{\int_0^zf(z')\mathrm{d}z'+\int_0^zg(z')\mathrm{d}z'},$$ where all functions are real and positive and $a>0$. Is there any ...
0
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2answers
68 views

Evaluate $\iint { \sqrt{\left| y-{ x }^{ 2 } \right|}\, dx\,dy } $ over a rectangle

Question: I want to evaluate $\iint_R {\sqrt{ \left| y-{ x }^{ 2 } \right|}\, dx\,dy }, $ where $R=[-1,1]\times[0,2]$. Indeed $x\in[-1,1]$ and $y\in[0,2]$. My approach: Since, $|y-x^2|$ is positive ...
0
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1answer
34 views

How this integral is evaluated $\frac{\partial }{\partial x}\left(\int _y^x\cos \left(-5t^2-2t-4\right)\:dt\right)$?

How this integral is evaluated? $$\frac{\partial }{\partial y}\left(\int _y^x\cos \left(-5t^2-2t-4\right)\:dt\right)$$ And in general, are there general methods for partial differentiation ...
0
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0answers
10 views

A Lebesgue integral with a $0$ denominator

Let $V\subset\mathbb{R}^3$ be an infinitely high solid cylinder, or a cylindrical shell of radii $R_1<R_2$, whose axis has the direction of the unit vector $\mathbf{k}$. For any point of ...
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1answer
27 views

Definite integral of trigonometric functions with complicated arguments

I came across beautiful integral (where $n$ is integer) $I(n, z) = \int_0^{\pi} \cos(nx) \sin(z \cos(x) ) \mathrm{d}x $ According to Gradshteyn and Ryzhik (p 414, Sec. 3.715, Eq. 13), solution is ...
2
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2answers
67 views

How much velocity can a canister of fuel give a spaceship?

I've recently considered the issue of how much velocity a canister of fuel can provide a 'spaceship'. I assumed we could approximate a basic solution If we know the mass of the fuel $m$, the mass of ...
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0answers
25 views

How to integrate discrete data by Gaussian quadrature method

I'm trying to numerically integrate discrete data by Gaussian quadrature method. The file attached test.mat is a discrete data set taken from a finite-element mode ...
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2answers
78 views

How to understand this integral result?

I was reading this page on Wikipedia: Birthday Attack I can understand up until how to approximate the minimal number of attempts for a given probability $$n(p; H) \approx \sqrt{2H \log \frac 1{1-p}...
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2answers
144 views

Evaluating $\int_0^{\infty} {\frac{\sin{x}\sin{2x}\sin{3x}\cdots\sin{nx}\sin{n^2x}}{x^{n+1}}}\ dx$

How to calculate $$ \int_{0}^{\infty}{\sin\left(x\right)\sin\left(2x\right)\sin\left(3x\right)\ldots \sin\left(nx\right)\sin\left(n^{2}x\right) \over x^{n + 1}}\,\mathrm{d}x $$ I believe that we ...
4
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2answers
114 views

Compute $\int\limits_Q^1\sqrt{(1-x^2)(1-\frac{Q^2}{x^2})}\mathrm{d}x$

I am trying to compute the integral $$\int\limits_Q^1\sqrt{(1-x^2)(1-\frac{Q^2}{x^2})}\mathrm{d}x$$ where $0\leq Q <1$ is a real number. I tried to substitute $x=\cos y,$ but this didn't bring ...
10
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3answers
230 views

How to prove that$\int_{0}^{1}\ln{(x/(1-x))}\ln{(1+x-x^2)}\frac{dx}{x}=-\frac{2}{5}\zeta{(3)}$

$$\int_{0}^{1}\ln{\big(\frac{x}{1-x}\big)}\ln{(1+x-x^2)}\frac{dx}{x}=-\frac{2}{5}\zeta{(3)}$$ Put $$\frac{x}{1-x}=y$$ $$I=\int_{0}^{\infty}\ln{y}\ln{(1+3y+y^2)}\frac{dy}{y(y+1)}=\frac{8}{5}\zeta{(3)...
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3answers
63 views

Choosing integration strategy [duplicate]

I need to integrate this function: $$\int \frac {x^3}{\sqrt{4-x^2}} \;dx$$ I don't know how i should integrate, should i divide because the numerator has an higher grade? There is any strategy i ...
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1answer
58 views

Is it true that $\int{\frac{\partial}{\partial t}f(x,t)\,\mathrm dx} = \frac{d}{dt}\int{f(x,t)\,\mathrm dx}$?

I was wondering why and when is true that: $\displaystyle\int{\dfrac{\partial}{\partial t}f(x,t)\,\mathrm dx} = \dfrac{d}{dt}\int{f(x,t)\,\mathrm dx}.$ I would appreciate any help.
2
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1answer
97 views

hard integral problems to solve

I'm practicing harder integration using techniques of solving with special functions I have difficulties with these two hard integrals; don't even know how to start, $$\int_0 ^\infty x^p e^{-\frac{\...
2
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0answers
30 views

Maximizing $\frac{\int_r^1xf(x)dx}{2-F(r)}$

Consider a continuous distribution on $(0,1)$ with probability distribution function $f$ and cumulative distribution function $F$. Define $$g(r)=\frac{\int_r^1xf(x)dx}{2-F(r)}$$ and let $r_M\in(0,1)$ ...
17
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10answers
2k views

What does it mean when dx is put on the start in an integral? [duplicate]

I have seen something like this before: $\int \frac{dx}{(e+1)^2}$. This is apparently another way to write $\int \frac{1}{(e+1)^2}dx$. However, considering this statement: $\int\frac{du}{(u-1)u^2} = \...
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1answer
20 views

Solving quasilinear PDE - 1D, time-dependant, convection

I have a task to solve the following quasilinear PDE (find $c(x,t)$): $$ c_x v + c_t = - v_x c $$ $c \in (0,20) , t \in (0, \infty)$ where I know function $v(x)$ to be $v(x) = \frac{3}{40}(1+\cos(\...
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1answer
407 views

limits of an integrable function over increasing sequence of sets

Let $(E_n)_{n \geq 1}$ be an increasing sequence of sets such that $\bigcup_{n \geq 1} E_n = \Omega$. Then for every integrable function $f$ we have $$\lim_{n \rightarrow \infty} \int_{E_n} f d\mu = \...
1
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1answer
33 views

Evaluate $\iiint_V zdV$, V is volume bounded below by cone $x^2+y^2 = z^2$ and above by sphere$ x^2+y^2+z^2=1$,lying on positive side of y-axis.

$\iiint_V zdV = \iiint_{\sqrt{x^2+y^2}}^{\sqrt{1-x^2-y^2}} zdzdydx = \iint_D \frac{1-2(x^2+y^2)}{2}dxdy$ where D is given by the disc $x^2+y^2 = \frac{1}{2}$ Changing x,y into cylindrical coordinates,...
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0answers
24 views

Integral over domain with implicit boundary

I have a unit disc (circle) $$x^2 + y^2 = 1$$ and a function $$\sum_{i=0}^N c_i\left(\sqrt{(x-a_i)^2 + (y-b_i)^2}\right)^3 = 0$$ I want to compute an area bounded by these two functions for $x$ ...
22
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3answers
1k views

Does integration by parts with “deja vu” have a name?

In some integration by parts problems, such as evaluating the integral of $e^x \cos x$ or $\sec^ 3 x$, one performs integration by parts (possibly more than once, and possibly together with algebraic ...
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1answer
15 views

Infinities on null sets

This is a conceptual question! Why is it that (e.g.) $\int_0^1 \frac{1}{x} dx$ doesn't converge. I'm stuck in the following way of thinking about it: Since the problematic part is $\int_0^\epsilon \...
4
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2answers
87 views

How can I solve this triple integral $\iiint_{B} y\;dxdydz$ on a defined set?

Calculate $$\iiint_{B} y\;dxdydz.$$ The set is $\;B=\{(x,y,z) \in \mathbb R^3$; $\; x^2+y^2+4z^2\le12$, $-x^2+y^2+4z^2\le6$, $y\ge 0 \}$. I know that B is defined by a real ellipsoid, an ...
0
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0answers
17 views

Volume of the $N$-dimensional domain $\sum\limits_{k=1}^N (1 + |x_k|^a)^b\le\epsilon$

I wish to calculate the following $N$-dimensional integral $$I = \int_0^\infty dx_1 \ldots \int_0^\infty dx_{N} \, H\left(\epsilon - \sum_{k=1}^N (1 + x_k^a)^b\right),$$ where $a, b$ and $\epsilon$ ...
0
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1answer
27 views

Integral substitution

I don't understand why the integral boundary change here from $[0,1]$ to $[0,\infty]$ $$\int_0^1 \int_{0}^\infty xe^{-x}f(ux,(1-u)x)\mathrm{d}u \mathrm{d}x$$ Substitution: $(ux=t,\ (1-u)x=s)\implies ...
4
votes
2answers
182 views

How to integrate $\int \dfrac{x^{13}\ dx}{x^5 + 1}$

We get this problem from our teacher today. I only wish that it was $x^{14}$ in the numerator, so we can use substitution method: $$\int \dfrac{x^{13}\ dx}{x^5 + 1}$$ I cant find way to integrate ...
3
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1answer
23 views

Converting Ellipse Integration Boundaries To Cylindrical Coordinates

I'm having the following integral, and I'm being asked to convert the integration boundaries to cylindrical coordinates. I've figured out that on XY-plane it's an ellipse having the following ...
11
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3answers
456 views

Meaning of $\int\mathop{}\!\mathrm{d}^4x$

What the following formula mean? $$\int\mathop{}\!\mathrm{d}^4x$$ I know that this $\int f(x)\mathop{}\!\mathrm{d}x$ is the integral of the function $f$ over the $x$ variable, but the following $\...
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1answer
70 views

Is there a reduction formula for $I_n=\displaystyle\int_{0}^{n\pi}\frac{\sin x}{1+x}\,dx$?

I haven't been able to manipulate this integral. I need to find the value of $I_n$ for $n=1,2,3,4$ and arrange them in ascending order.
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0answers
13 views

Correlation of an integral

Assuming that $a(x)$ is a random variable and correlation $\xi(x_1,x_2)=\langle a(x_1)a(x_2)\rangle$ is known (where angular bracket denote statistical averaging), it it possible to write the ...
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4answers
110 views

Find $\int_0^{\pi}\sin^2x\cos^4x\hspace{1mm}dx$

Find $\int_0^{\pi}\sin^2x\cos^4x\hspace{1mm}dx$ $ $ This appears to be an easy problem, but it is consuming a lot of time, I am wondering if an easy way is possible. WHAT I DID : Wrote this as $\...
3
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2answers
41 views

Application Banach-Alaoglu Theorem

When reading about Banach-Alaoglu Theorem on Wikipedia, I read the following assertion: '' Let $f_n$ be a bounded sequence of functions in $L^p$. Then there exists a subsequence $f_{n_k}$ and an $f\...
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0answers
50 views

$\lim_{n \to \infty}\int_{0}^{a} e^{nx} dx$

Find $\lim_{n \to \infty}\int_{0}^{a} e^{nx} dx$ This seems to be a straightforward problem but since I am new to defnite integrals and thishas appeared in one the graduate exam papers I am looking ...
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0answers
46 views

Simplify an integral or the integrand involving hyperbolic functions

I would like to simplify the following integral or at least the integrand: $$f(t):=\int_{a}^{t-a} \frac{(\cosh(t-x)-\cosh(a))^{i\tau-1/2}}{(\cosh(x)-\cosh(a) )^{i\tau+1/2}} dx, \quad t>2a,$$ ...
1
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0answers
46 views

Why does Maple include x in the solution of this definite integral?

I have the following function defined in Maple: $$ f(x) := (2 - a + ax^2) \sqrt{1 + 4a^2x^2} $$ And I want to calculate the definite integral of this from -1 to 1: $$ \int_{-1}^{1}{f(x)dx} $$ I do ...
32
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1answer
794 views

Weber-type integral

In connection with this answer, I came across the following integral: $$\int_{0}^{\infty} \frac{du}{u} \: \,e^{-t u^2} \frac{J_0(u) Y_0(r u)-J_0(r u) Y_0(u)}{J_0^2(u)+Y_0^2(u)}$$ where $r \gt 1$. I ...
0
votes
1answer
2k views

Integrating a solid using cartesian, cylindrical and spherical coordinates

The region $W$ is the cone shown below (see image). The angle at the vertex is $π/3$, and the top is flat and at a height of $7\sqrt{3}$. Write the limits of integration for $\int_W dV$ in the ...
2
votes
1answer
111 views

On a solution to a triple integral.

I want to calculate the function $f(x,y,z) = z$ on the set $B = \{ (x,y,z) \in R^3 | z \ge \sqrt{7x^2 + 3y^2}, 2x + z \le 3 \}$ I tried to solve it without cylindrical substitutions. the solution is ...
0
votes
1answer
73 views

Find the limit $\lim_{x\to 0} x^{-3}\int_0^{x^2}\sin{(\sqrt t)}dt$

I use the fundemental theorem of calculus $$ \displaystyle\lim_{x\to 0}\frac{\displaystyle\int_0^{x^2}\sin{(\sqrt t)}dt}{x^3}=\frac{F_{(x^2)}-F_{(0)}}{x^3}="\frac{0}{0}" $$ Than apply L'hopital rule ...
4
votes
1answer
46 views

Conditions on a complex measure to be real

Let $(X,\mathcal{S}, \mu)$ be a measure space with $X$ a locally compact Hausdorff space, $\mathcal{S}$ the Borel subsets of $X$ and $\mu$ a complex measure. Suppose that $$ \int_X f \ d\mu \in \...
0
votes
0answers
42 views

How to calculate this integral $\int_0^a {dx\frac{{\tanh \left( x \right)}}{x}} $? [duplicate]

I encountered this integral, which Mathematica can't give an answer. $$\int_0^a {dx\frac{{\tanh \left( x \right)}}{x}} $$ I am sure the result contains Euler constant. How to do it?
2
votes
2answers
170 views

Nasty double integral with lots of exponentials

I am trying to compute a double integral. I will first define the functions that make up the integrand: $$F(\gamma)= A \,\exp(-a \, \gamma^ {1/2})+B \, \gamma^{-1/2}\left(1-\exp(-b\gamma^ {1/4})\...