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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48 views

When would you want to model a derivative?

I just read this, and am intrigued. http://formulize.nutonian.com/documentation/eureqa/tutorials/modeling-derivatives/ What kind of model would you have a scatter plot of data points, and want to ...
0
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3answers
77 views

Evaluate $\int\cos^3x\sin^2x\,dx$

My university mathematics is kind of poor. But as I am learning advanced mathematics this becomes a major shortcoming. I tried to integrate $\int$$\sin^3\theta$$\cos^2\theta$d$\theta$ = ...
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2answers
145 views

How to evaluate $\int _{-\infty }^{\infty }\!{\frac {\cos \left( x \right) }{{x}^{4}+1}}{dx}$

How to evaluate the following integral? $$ \int _{-\infty }^{\infty }\!{\frac {\cos \left( x \right) }{{x}^{4}+1}}{dx} $$ Unlike this example, according to maple, the solution does not contain sine ...
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2answers
60 views

Integrating of product of sine and cosine

$$ \int_L^{-L} \cos{\frac {n \pi x}{L}} \sin{\frac {m \pi x}{L}} dx = 0 $$ for any $n$ and $m$ . I did integrating by parts, but I am not getting this equals to $0$. Can anyone show this?
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0answers
55 views

integrate over quadrant of sphere

It is known that $\int_{x\in S}\exp(\kappa\mu^Tx)dx$ where S is the surface of the unit sphere is $\frac{(2\pi)^{p/2} I_{p/2-1}(\kappa) }{\kappa^{p/2-1}}$ where $p$ is the number of dimensions and $I$ ...
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1answer
33 views

What is the joint CDF of f(x,y)=2(x+y) 0<=x<=y<=1

I am trying to find the joint CDF of $f(x,y)=2(x+y) : 0\leq x\leq y\leq 1$. There are five different answers for the CDF depending on the restrictions of $x$ and $y$ that you use. I found the CDF ...
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0answers
152 views

Inclined Elliptic Tank Volume Calculation

Can someone help me determine an equation for calculating the volume of an elliptical cylinder on it's side and inclined 5 degrees from horizontal? The tank has flat ends. I have found several ...
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1answer
41 views

What algorithm to solve this integral similar to a normal CDF numerically?

I'm looking to solve this integral numerically. It is a bit similar to a normal CDF. z and tau are deterministic. What kind of algorithm may do the job, ideally to be coded in C/C++? I'm ...
1
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1answer
44 views

Finding the antiderivative of a real power of a rational function

my abilites in integration (applied) are very limited, so my question is: if i would like to find out how to approach a problem like finding the antiderivative of the function $f(x) = ...
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1answer
28 views

How to take the integral of a derivative to obtain desired result?

I am aiming for the form of derivative below computed over time that causes its differentiated variable V to go from an initial -.001 and increase to reach 10. I will explain my current calcs below ...
3
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1answer
86 views

Integrating tensors on manifolds

When/how can you integrate tensors on manifolds and what does it mean? I imagine that line integrals of tensors make sense when you have a connection, since you can uniquely parallel transport all ...
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4answers
84 views

$\int \sqrt{x} 2^{-\sqrt{x}}dx$. How to begin?

Let's take: $$\int \sqrt{x} 2^{-\sqrt{x}}dx$$ I don't know how to begin, I am asking for advice.
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2answers
63 views

How to determine this integral [duplicate]

Let's take: $$\int \frac{1}{x\sqrt{x+1}} \ dx $$ I tried solve this by four hour, so I am asking for help
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0answers
30 views

Contour Integration Problem for reciprocal of Bessel Function Power

Hopefully some one can help me with this problem. I have to compute $$ \oint_C \frac{1}{[J_0(\alpha t)]^n}dt$$ where $C$ is a contour that contains the first zero ...
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2answers
73 views

What is the difference between $\delta x$ and $dx$

Sometimes I see the $\delta x$ and $dx$ but I don't know exactly what is the difference between them.
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1answer
114 views

Find the closed form of the digamma related series

The question I asked here Computing $\sum_{n=1}^{\infty} \left(\psi^{(0)}\left(\frac{1+n}{2}\right)-\psi^{(0)}\left(\frac{n}{2}\right)-\frac{1}{n}\right)$ made me think to ask for your support for ...
4
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1answer
135 views

Contour integration of $\int_{-\infty}^{\infty}\frac {\sin^3 x}{x^3} dx$: where are the singularities?

I have just begun to study complex analysis and I'm trying to calculate $$ \int_{- \infty}^{\infty} \frac {\sin^3 x}{x^3} dx $$ with the "help" of an exercisebook. I have followed these steps: ...
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0answers
47 views

Calculus question table leg

Calculate the volume of the “table leg” for which the radius r at height z is given by the following expression, with $0 \leq z \leq 12 $: $$r(z)=4+\cos \left( \frac{\pi z}{2} \right)$$ then it ...
0
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3answers
67 views

Evaluating $\int_{0}^{1} \frac{1}{(1+x^2)^{5/2}}dx $

$$\int_{0}^{1} \frac{1}{(1+x^2)^{5/2}}dx $$ it says to let $x = \tan(u)$ which i presume wants $$\frac{dx}{du} = \sec^2(u) = $$ $$dx = \sec^2(u) du $$ $$\int_{0}^{1} ...
0
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1answer
40 views

Evaluate $\displaystyle\iiint_W x^2 \cos z\,dv$

$$\displaystyle\iiint_{W}{x^2\cos z \ dv}$$ Where $W$ is the region bounded by $z=0, z=\pi, y=0, y=1,$ and $x+y=1$. I drew the region $W$ at home and found that it is a uniform triangular prism of ...
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1answer
46 views

Relation between sum and integral

I have an exercise (from physics) where I am supposed to show $$\sum_{k'<k_f} \frac{1}{|k-k'|^2} = C \left( \frac{1}{2} + \frac{1-(\frac{k}{k_f})^2}{4 \left( \frac{k}{k_f} \right) } ln |\frac{1 + ...
1
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2answers
49 views

Integral Measures: Variation

Given a measure $\lambda\geq0$. Regard a real function $h:\Omega\to\mathbb{R}$ with $h\in\mathcal{L}$. Consider the real measure $\mu(E):=\int_E h\mathrm{d}\lambda$. Then its total variation ...
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0answers
27 views

Transformation theorem for triple integrals

Let $U = [0, 1] \times [1, 2] \times [0, 1]$, and $V = G(U)$. Find $V$, and write the triple integral $$ \exp(-x_3^2)\sin(x_1x_2) \,dx_1\,dx_2\,dx_3$$ as an integral in terms of the ...
3
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3answers
78 views

Question about an integral.. Why is $a = $2?

I found the item in monbukagakusho 2013 math B exam. Consider the function $$F(x) = \int_a^x f(t)\ dt = x^3 - 2x^2 - x - a$$ with $a \ne 0$. Find $a$. I looked at the answer sheet and $a = ...
5
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2answers
149 views

Showing that $ \int_0^1 x^{2n}f(x) dx = 0 $ implies $f = 0$

This is my question: Show that if $f \in C[0,1]$ satisfies $ \int_0^1 x^{2n}f(x) dx = 0 $, then $f$ is the zero function. Note: I am aware that a similar question to this has been asked on maths ...
1
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2answers
25 views

Volume of rotated region (integration)

Let $T$ be a right-angled triangular region with vertices $(0,−b)$,$(1,0)$ and $(0,a)$ where $a$ and $b$ are positive numbers. When $T$ is rotated about the line $x=2$, it generates a solid with ...
1
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0answers
39 views

How to find the limits of this integral?

I want to prove the stoke's theorem given $\int \int _{\sigma }\left[\nabla \:\times \:\left(y\:\hat{i}+2\:\hat{j}\right)\right]\cdot \hat{n}\:d\sigma $, where $\sigma $ is the surface in the first ...
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3answers
526 views

How to integrate the dilogarithms?

$\def\Li{\operatorname{Li}}$ How can you integrate $\Li_2$? I tried from $0 \to 1$ $\displaystyle \int_{0}^{1} \Li_2(z) \,dz = \sum_{n=1}^{\infty} \frac{1}{n^2(n+1)}$ $$\frac{An + B}{n^2} + ...
3
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1answer
60 views

$\int_{a-\epsilon}^{a+\epsilon} \delta(x - a)dx = 1$

$$\int_{a-\epsilon}^{a+\epsilon} \delta(x - a)dx = 1$$ I can see intuitively why this is so as $a$ is inside the the domain of integration and all other values in the domain contribute $0$ to the ...
5
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4answers
89 views

Evaluate $\int t^2 e^{-2i\pi nt}\,dt$

I need to get $$\int t^2 e^{-2i\pi nt}\,dt$$ I'm thinking to use integration by parts, but $\int e^{-2i\pi nt}\,dt$ is tripping me up. Can anybody help? Thanks!
0
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1answer
111 views

Rieman-Stieltjes integration “by parts”

Let us define a partition $a=x_0<x_1<...<x_n=b$ of interval $[a,b]$ and let us define the Riemann-Stieltjes integral $\int_a^b fdg$ of a function $f:[a,b]\to\mathbb{C}$, or ...
13
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1answer
261 views

Integral that arises from the derivation of Kummer's Fourier expansion of $\ln{\Gamma(x)}$

I am trying to prove that for $0<x<1$, $$\color{blue}{\ln{\Gamma(x)}=\frac{1}{2}\ln(2\pi)+\sum^\infty_{n=1}\left\{\frac{1}{2n}\cos(2\pi nx)+\frac{\gamma+\ln(2\pi n)}{n\pi}\sin(2\pi ...
1
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0answers
122 views

Need help finding the volume of the solid of revolution

I need finding the volume of the solid of revolution (a cup), convert the volume to ounces and having the calculation be a close approximate to the measured ounces. I'm using the equation of the line ...
0
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1answer
51 views

Riemann sums with equidistant sample points converge to the integral

Suppose $f \in R(x)$ on $[0,1]$. Define $$a_n = \frac 1n \sum_{k=1}^n f(\frac kn)$$ for all n. Prove $\{a_n\}_{n=1}^\infty$ converges to $\int_0^1 fdt$. Honestly I've been trying to tackle this ...
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0answers
23 views

What is a function in $f\in L^n(\mathbb{R}^n)$ but $g(x)=\int_{|y|<1} \frac{|f(y)|}{|x-y|^{n-1}}dy$ is not in $L^\infty$

What is a function in $f\in L^n(\mathbb{R}^n)$ but $$g(x)=\int_{|y|<1} \frac{|f(y)|}{|x-y|^{n-1}}dy$$ is not in $L^\infty$. I have no idea where to start. Apparantly this is related to the ...
1
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1answer
30 views

Integration $\int\frac{\sqrt{x+4}}x dx$ by partial fraction

Here's what I came up with: $$\int\frac{\sqrt{x+4}}x dx$$ for $ u = \sqrt { x + 4 } $ $$=\int\frac{u}{u^2-4}\;2u\;dx$$ $$=2\int\frac{u^2}{(u-2)(u+2)}\;dx$$ $$=\frac{A}{u-2}\;+\;\frac{B}{u+2}$$ ...
1
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1answer
45 views

Change of Variables for double integration

I'm doing the following exercise. However I don't get the book's result: Consider the transformation $T$ given by the equations: $$x=u+v,\;\;\;\;\;\;y=v-u^2.$$ A triangle Q in the plane $(u,v)$ has ...
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1answer
26 views

If $\lvert \int fa \rvert, \lvert \int gb \rvert \geq \alpha$, what about $\lvert \int fgab \rvert+\lvert \int fgb \rvert+\lvert \int fga \rvert$?

In the course of a project, I'd like to prove or disprove the following statement: let $f,g$ and $a,b$ be real-valued functions defined say on $\mathbb{R}$, and $\alpha > 0$ such that $\lvert ...
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5answers
1k views

Humorous integration example?

I was just reading though an introductory calculus book and it has the note: NOTE When integrating quotients, do not integrate the numerator and denominator separately. This is no more valid in ...
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1answer
25 views

$\int_{\mathbb{R}^2} (z-ax^2-by^2)^n x^2 \chi_{[0,z]}(ax^2+by^2) \, dx \, dy, $

I want to calculate $$\int_{\mathbb{R}^2} (z-ax^2-by^2)^n x^2 \chi_{[0,z]}(ax^2+by^2) \, dx \, dy, $$ where $z>0, a>0, b>0$ and $\chi$ is the characteristic function that is one if ...
1
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2answers
49 views

$\int_{\mathbb{R}^2} \delta(E-ax-by) x^2 dx $

I am wondering how we have to integrate $\int_{\mathbb{R}^2} \delta(E-ax^2-by^2) x^2 dxdy.$ I am not familiar with this kind of delta distribution (depending on two coordinates), so I was wondering if ...
4
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1answer
35 views

Proving that $(2 \pi i)^{-1} \int e(\pi v^2/y^2) x^v y^{-1} dv = e(-\pi (\log x)^2 y^2 /4)$

I've seen a particular integral transform (an inverse Mellin Transform) used a few times, but I don't know how it's proved. In particular, I'm trying to prove $$\frac{1}{2\pi i} \int_{(2)} e^{\pi ...
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1answer
48 views

Convergence of $\int_1^{+\infty}\frac{\sin(x)}{x}\arctan(x) \mathop{d}x$

How to prove that integral $$\int_1^{+\infty}\frac{\sin(x)}{x}\arctan(x) \mathop{d}x$$ converges (or does not)?
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1answer
41 views

Integral equal to 0'th Bessel function

Mathematica tells me $\int_0^{2\pi}e^{-ikr(\cos\theta_k\cos\theta+\sin\theta_k\sin\theta)}d\theta=2\pi J_0(kr)\ ,$ where $0<k\in\mathcal{R}$, $0\leq r\in\mathcal{R}$, $\theta_k\in\mathcal{R}$ and ...
0
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1answer
37 views

Calculate this integral in $N$-dimensional space

I want to calculate the integral $$\int_{\mathbb{R}^N \times \mathbb{R}^N} \chi_{[0,E]}\left(\sum_{i=1}^N \frac{p_i^2}{2m} + \frac{m \omega^2 q_i^2}{2} \right) \,dp\, dq.$$ Now I should explain what ...
1
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1answer
35 views

Convergence of a sequence of functions integrated over a sequence of measures

I have real-valued functions $\{f_n\},f$ on a subset $X\subset \mathbb R^n$ that are equicontinuous and I have Borel measures $\{\mu_n\},\mu$. I have that For each fixed $m$, $\int f_m d\mu_n\to\int ...
1
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2answers
96 views

What exactly is a vector line integral?

I am having trouble wrapping my head around the idea and understanding what is actually happening, and I am hoping for a more intuitive explanation, or at least a better understanding of what is ...
1
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1answer
65 views

Riemann-Stieltjes Integral and the Step Function

Let $a < c< b$ and let $\alpha (x)$ be defined as $\alpha (x) =\begin{cases} 0 & a \le x \le c \\ 1 &c<x \le b \end{cases}$. Show that $f \in \mathcal{R}(\alpha)$ if and only if ...
4
votes
2answers
55 views

Better substitution calculating integral?

I'm calculating $$ \iint\limits_S \, \left(\frac{1-\frac{x^2}{a^2}-\frac{y^2}{b^2}}{1+\frac{x^2}{a^2}+\frac{y^2}{b^2}} \right)^\frac{1}{2} \, dA$$ with $$S =\left\{ (x, \, y) \in \mathbb{R}^2 : ...
0
votes
0answers
35 views

Limit and integral exchange when the range of integral includes the variable of the limit.

$\lim_{N->\infty}\int_0^{N\pi}(\frac{\sin(x)}{N\cdot \sin(x/N)})^2dx$ My question is how to simplify the above expression. My naive thought is ...