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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0
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0answers
9 views

rectangular contour

I am trying to do the inverse transform of $$F(k) = \frac{1}{(i*\sinh{ak})}$$ I have to do a rectangular contour with the top at y = pi/a But in the solution book they move the contour down so it ...
0
votes
1answer
33 views

Summation of $\int\frac{1}{1+x}dx$ in a range of 1 to infinity. [closed]

Let $0 < \alpha < \beta < 1$. Then $$\sum_{k=1}^\infty \int_{1/(k+\beta)}^{1/(k+\alpha)} \frac{1}{1+x}dx$$ is equal to $$ \begin{align} &(A)\ln \frac{\beta}{\alpha}\qquad\qquad ...
0
votes
0answers
13 views

application of integration in graphics

Is Integration used in animation/computer graphics? If yes, then how it is used. A couple of example would be great . Thanks
0
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0answers
27 views

On bounding the integral $\int_{x_0}^x te^{-t \ln t + t -1} dt$.

I was following a lecture in class and the Professor was trying to figure out what happened to the following integral $$\int_{x_0}^x te^{-t \ln t + t -1} dt$$ When $x$ got very large or very low. He ...
2
votes
1answer
84 views

Volume from equation $(x ^2+ y ^2 + z ^2 ) ^2 = xyz$

How can you calculate the volume of the shape represented by the following equation: $$(x ^2+ y ^2 + z ^2 ) ^2 = xyz$$ I tried converting it to polar form (so $r = ...
1
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0answers
34 views

How to conclude converges of an integral if I know that $\lim _{x\to \infty }e^{2x}f\left(x\right)=3$ and $f$ continuous and bounded in $\mathbb{R}$?

Let $f(x)$ be continuous and bounded at $\mathbb{R}$. And $\lim _{x\to \infty }e^{2x}f\left(x\right)=3$ prove that $\int _0^\infty\:f\left(\ln x\right)dx$ converge. What I know is: 1)there's a ...
2
votes
1answer
56 views

Finding a function based on its Derivative without Integrating

My question revolves around finding a function based on its derivative of the type below : Problem : The limit below represents the derivative of some real-valued function $f$ at some real-number ...
-1
votes
2answers
55 views

Quantum mechanics. [closed]

If $P(x) =Axe^{-x^2/a^2}$ for $x > 0$ and $P(x) = 0$ for $x < 0$, find $A$ such that $$\int_{-\infty}^{\infty}P(x)dx=1$$ And hence calculated the expected value of $\left\langle x\right\rangle$ ...
1
vote
1answer
38 views

Integral similar to fresnel integrals

$$\int_{0}^{+\infty} \frac{e^{-r^2}}{r^2-i\gamma^2} dr = ?$$ I tried the normal semicircular contour integrals, but there is always a problem with the exponential when I close the contour. This post ...
0
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0answers
10 views

Slowly increasing Gauss Hermite rule

I am new to using Sparsegrids and am trying to understand how 'slowly increasing Gauss Hermite' points are generated? Most reference points towards: Book chapter(4) (See Table 4.1) or a thesis. (see ...
-1
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0answers
17 views

bessel function integration [on hold]

$$ I(u,v_p) = \int_{0}^{{\it v_p}}\! \left| \int_{0}^{1}\!{{\rm e}^{iu{\rho}^ {2}}}{{\rm J}_{0}\left(v\rho\right)}\rho\,{\rm d}\rho \right| ^{2}v\,{\rm d}v $$ Suppose: P=1 and J is the ...
0
votes
0answers
25 views

Proof that $f[x_0,x_1,…,x_n,\epsilon,\epsilon]=\frac{f^{n+2)}(\eta)}{(n+2)!}$

Up to now i have the following rule for divided differences: Assuming $x_0 \le x_1 \le...\le x_n$ then If $x_0 \lt x_n$ then ...
-1
votes
2answers
70 views

How to integrate $\int \frac{1}{\sqrt{(1+x^2)^3}} dx$ with substitution:$ x = tan(\alpha)$ [closed]

$$\int \frac{1}{\sqrt{(1+x^2)^3}} dx$$ with substitution:$$ x = \tan(\alpha)$$
0
votes
1answer
24 views

integral of two functions absolute

I've got the two function: f(x) = -4x + x³ and g(x) = 5x they meet each other at -3, 0 and 3, where the areas between -3 and 0 ...
0
votes
1answer
68 views

Compute the Lebesgue integral $\int_0^{\infty} \frac{x}{e^x -1}dx$.

Compute the Lebesgue integral $\int_0^{\infty} \frac{x}{e^x -1}dx$. I think I need to use the Dominated Convergence Theorem or the Beppo Levi Theorem to show this, but I don't really know what I ...
0
votes
0answers
17 views

Find volume and surface of a body [closed]

Find volume and surface of a body $$G=\{(x,y,z)\in\mathbb R^3|2x\le x^2+y^2\le1,-\sqrt{ x^2+y^2}\le z\le4- x^2-y^2, y\ge0\}$$ Some directions will be helpfull.
2
votes
1answer
54 views

$f: \mathbb{R} \to \mathbb{R}$ is Lebesgue integrable. Does it follow that $\lim_{x\to \infty} f(x)=0$?

Suppose that $f: \mathbb{R} \to \mathbb{R}$ is Lebesgue integrable. Does it follow that $\lim_{x\to \infty} f(x)=0$? What if $f$ is continuous on $\mathbb{R}$? I think the first question is false but ...
1
vote
0answers
20 views

Volume between $z=13-x^2-y^2$ and $z=4 \sqrt{x^2+y^2} + 1$ Polar Integration

Find volume between $z=13-x^2-y^2$ and $z=4 \sqrt{x^2+y^2} + 1$ My attempt: Convert to polar, and find intersections. $r^2=x^2+y^2$, giving $z=13-r^2$ and $z=4r+1$ This means $r=-6, 2$ I just ...
1
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0answers
26 views

How would I differentiate an integral with bounds?

Let $\space f \space$ be a differentiable function of $\space x \space$. Now I know that for the following integral: $$I=\int f(x) \space dx$$ Clearly: $${dI\over dx}=f(x)$$ Since integration is ...
7
votes
2answers
69 views

Evaluation of $\int_{0}^{10 \pi} ([\sec ^{-1}x]+[\cot^{-1} x])~\mathrm dx$ [closed]

Find the value of the integral $$\int_{0}^{10 \pi} (\lfloor\sec ^{-1}x\rfloor+\lfloor\cot^{-1} x\rfloor)~\mathrm dx$$ where $\lfloor . \rfloor$ denotes greatest integer function. Could some help me ...
0
votes
1answer
9 views

$x^2+y^2\le 1$; $z=\sqrt{x^2+y^2}$; and $x^2+y^2=4-z$

I need to find a value and "surface" of a body which is contained in the following contours: $x^2+y^2\le 1$; $z=\sqrt{x^2+y^2}$; and $x^2+y^2=4-z$. Some hints and directions will be helpful. Sorry for ...
0
votes
0answers
37 views

Volume enclosed by Implicit Surface

I am trying to calculate the Volume that's enlcosed by the surface: $$(x^2 + y^2 + z^2)^2 = xyz$$ The following is what i tried. I rewrote it in spherical coordinates where $x=r \cdot \sin\vartheta ...
2
votes
2answers
38 views

Simple integration problem: $h\int_0^T (Q-at)dt$

I'm not a math student, but I need to use the following formula for my project. I have an equation: $$h\int_0^T (Q-at)dt$$ Also, $Q=aT$ (not sure would this help). I want to integrate this ...
2
votes
1answer
36 views

Modulus of roots of polynomial tend to infinity

Define $f_n:\mathbb{C}\to\mathbb{C}$ and $(\alpha_n)$ such that:$$f_n(z)=\sum_{k=0}^n \frac{z^k}{k!}$$ and $f_n(\alpha_n)=0$. Prove $|\alpha_n|\to\infty$ as $n\to\infty$. I guess this makes sense ...
1
vote
3answers
91 views

Integral of $\frac{x}{(1-x^3)\sqrt{1-x^2}}$

In the pool of difficult (at least to me) integrals I've been trying to solve this one: $$\int\frac{x}{(1-x^3)\sqrt{1-x^2}}dx$$ Since Wolfram Alpha has been helpful with all the other integrals ( at ...
1
vote
2answers
33 views

Analysis of $(y-x)^2 =x^3 $

I was doing some tasks in integral application and came across this one: Calculate the surface area bounded by $(y-x)^2 = x^3$ and line $x=2$ I started doing this the usual way, when I realized ...
1
vote
2answers
55 views

If $f$ continuous at $[1,\infty)$ and $\int _1^\infty\,f\left(x\right)dx$ converge, then $\int _1^\infty\frac{f\left(x\right)}{x}dx$ also converge?

I need to prove or disprove this statement. I didn't find any counter example. So I tried to prove it. I know that $\int _1^\infty\,f\left(x\right)dx$ converge so the limit $\lim _{t\to \infty }\int ...
2
votes
0answers
51 views

Integral with irrational functions and polynomials

I thought this integral was simple, but it turns out it's not. $$\int \frac{xdx}{\left(1-x^3\right)\sqrt{1-x^2}}$$ I tried the substitution $1-x^3=\frac{1}{t}$, but that leaves me again with $x = ...
2
votes
4answers
68 views

Solve for $\alpha$: $P = \frac{1}{\sigma}\displaystyle\int_{0}^{\alpha} \exp (\frac{ -2 x^{\beta}}{\sigma} ) dx$

I need to solve: $$P = \frac{1}{\sigma}\displaystyle\int_{0}^{\alpha} \exp ( \frac{ -2 x^{\beta}}{\sigma} ) \;dx $$ This simplifies to: $$P = \frac{1}{\sigma} \displaystyle\int_{0}^{\alpha} \exp (- ...
0
votes
1answer
74 views

Proving $\int_{-\infty}^{+\infty} {{\cos(mx)}\over(x^2+a^2)(x^2+b^2)}dx={\pi(ae^{-mb}-be^{-ma})\over ab(a^2-b^2)}$

Show that $$\int_{-\infty}^\infty {{\cos(mx)}\over(x^2+a^2)(x^2+b^2)}dx={\pi(ae^{-mb}-be^{-ma})\over ab(a^2-b^2)}$$ where $a,b,m>0$ and $a$ is not equal to $b$. I already know that ...
0
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0answers
15 views

A certain multidimensional integral.

Consider a following multidimensional integral: \begin{equation} \bar{I}^{(t_0,t)}_p := \int\limits_{t_0 \le \xi_0 \le \cdots \le \xi_{p-1} \le t} \prod\limits_{j=0}^p (\xi_{j-1}-\xi_j) \cdot ...
1
vote
0answers
28 views

Integral of a function $f: \mathbb{N}\times\mathbb{N} \to\mathbb{R}$

Let $X = Y = \mathbb{N}$, $A = B = P(\mathbb{N})$, $\mu$ and $\nu$ counting measures on $(X, A)$ and $(Y, B)$. Define $f:X\times Y \to \mathbb{R}$ by $f(m,m) = 1$, $f(m+1,m) = -1$ and $f(m,n) = 0$ ...
-1
votes
3answers
64 views

How to find the arc length of a curve? [closed]

Given: $x=4e^tcos(t)$ and $y=4e^tsin(t)$ and a $t$ range from $0$ to $\pi$. I can find $$\frac{dx}{dt}$$ and $$\frac{dy}{dt}$$ and setup the initial integral, but I am not sure where to go from there. ...
0
votes
0answers
40 views

need help to solve the following integral [closed]

I need help to solve the following integral. It seems bounded and well defined but I don't know how to solve it. I used series expansion of tanh[x] but then I got answer as series which I could not ...
-2
votes
1answer
34 views

Integrating division [closed]

I need to find out if following equation is integrable enter image description here I've tried and can't find an integration, but the formula is so simple that it must have one? Right?
0
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2answers
68 views

Solve for $\alpha$: $P = \int_{0}^{\alpha} \sqrt{\frac{\sqrt{6}}{8\pi\sigma_{x}|x|}} e^{\frac{-\sqrt{3} |x|}{\sqrt{2}\sigma_{x}}} dx$

So I need to solve the following for $\alpha$: $$P = \displaystyle\int_{0}^{\alpha} \sqrt{\frac{\sqrt{6}}{8\pi\sigma_{x}|x|}} \exp \left( \frac{-\sqrt{3} |x|}{\sqrt{2}\sigma_{x}} \right) dx$$ If I ...
2
votes
2answers
49 views

Evaluating integral with a singularity.

I want to evaluate an integral numerically that contains one singularity. The software I use for this is Python. The actual integral I want to evaluate is quite long with a lot of other constants so I ...
2
votes
0answers
78 views

how to solve this integral ? It seems bounded and well defined integral but I don't know how to solve this

how to solve the following integral ? It seems well defined i.e. bound but I could not solve it. I tried by expanding series expansion of tanh[x] but after that I got a series as an answer, which I ...
2
votes
1answer
62 views

Application of Fubini-Tonelli's Theorem on function $\frac{2}{\pi}e^{-ax}\cos(x\cos{\theta})$

The question asks me to prove that $$\int_0^\infty J(x)e^{-ax}dx=\frac{1}{\sqrt{1+a^2}},$$ where $a>0$ and $J(x)=\frac{2}{\pi}\int_0^{\pi/2}\cos(x\cos{\theta})d\theta.$ I started off by ...
3
votes
2answers
44 views

Contour Integration of $\sin^2(x)/(1+x^2)$

How should I calculate this integral $$\int\limits_{-\infty}^\infty\frac{\sin^2x}{(1+x^2)}\,dx\quad?$$ I have tried forming an indented semicircle in the upper half complex plane using the residue ...
0
votes
1answer
22 views

Function of bounded variation and integration

Let f belong to $C[a,b]$. Show that there is a function g that is of bounded variation on [a,b] for which $\int_a^bfdg=||f||_{max}$ and TV(f)=1. This problem appears on page 162 of Royden's Real ...
1
vote
0answers
14 views

derivative of 2 dimensional integral

Let $f: \mathbb{R}^3 \rightarrow \mathbb{R}: (x,y,t) \mapsto f(x,y,t)$ a derivable function in every direction. Define $\mathfrak R_{\alpha}(u,t) := \int_{L(\alpha,u)} f(x,y,t) d(x,y)$ met ...
1
vote
0answers
35 views

Help understanding integral proof of variational auto-encoder

I'm trying to understand the key proof of the parametrization trick of this paper "Auto-Encoding Variational Bayes" http://arxiv.org/pdf/1312.6114v10.pdf. It's section 2.4 on page 4. which states ...
0
votes
1answer
21 views

Probability for random vector given probability distribution [closed]

Given the following probability distribution: $f(x,y) = \begin{cases} xe^{-x-y}, & x,y>0 \\[2ex] 0, & \text{elsewhere} \end{cases}$ compute $P(X \le Y)$. I know that the result is $1/4$, ...
1
vote
2answers
98 views

Prove that $(n+1)\int_0^1{\frac{x^n}{x+1}dx}\to\frac12$ [duplicate]

Let $I_n=\int_0^1{\frac{x^n}{x+1}dx}$ , $n>0$ Show that $\lim_{n->\infty}{(n+1)I_n} = \frac{1}{2}$ All I could do was to show that the $I_n$ is decreasing.
14
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0answers
258 views

We have sums, series and integrals. What's next?

We know how to sum or average a finite number of terms: sums. We know how to sum a countable infinite number ${\beth_0}$ of terms: series. We know how to sum ${\beth_1}$ terms: integrals. How to ...
0
votes
2answers
63 views

Wolfram answer is different for the integral $\sqrt{\frac{x}{2-x}}dx$

$$I=\sqrt{\frac{x}{2-x}}dx=\int \frac{xdx}{\sqrt{2x-x^2}}=\frac{-1}{2} \times \int\frac{(2-2x-2)dx}{\sqrt{2x-x^2}}$$ so $$I=\frac{-1}{2}\int\frac{(2-2x)dx}{\sqrt{2x-x^2}}+\int ...
3
votes
3answers
83 views

Integrating $\int \frac{\sqrt{x^2-x+1}}{x^2}dx$

Evaluate $$I=\int\frac{\sqrt{x^2-x+1}}{x^2}dx$$ I first Rationalized the numerator and got as $$I=\int\frac{(x^2-x+1)dx}{x^2\sqrt{x^2-x+1}}$$ and splitting we get ...
0
votes
2answers
38 views

question about integration of symmetrical graph to find area

let's say, we have an symmetrical curve such as $x=\sqrt{y}$ If we integrate from 4 to 0, wouldn't it cancel out with the area on the other side of the axis? When integration is performed from 4 ...
2
votes
1answer
25 views

Integral of a Gradient function (and another function?)

I'm aware of line integrals around planes and curves, but I cannot make up how to approach this question: $$\int_C f∇f \cdot \,d\mathbf{r} $$ where $f(x,y,z)=xz\cos(x^2+y^2)$ and C is the ...