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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3
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1answer
39 views

Integration to Gamma Function?

I need to show that $$\int\limits_{0}^{\infty}\theta^{-\tau(\alpha_1 + \alpha_2) - 1}e^{-\theta^{-\tau}\left(y^{\tau} + \delta^{\tau}\right)}\text{ d}\theta = \dfrac{\Gamma\left(\alpha_1 + ...
0
votes
0answers
12 views

Probability density function of an element

How to find the probability density function of $x_m\left(1\le m\le n\right)$ from joint density function, $p_X\left(x_1,x_2,\cdots,x_n\right)$, of $n$ random variables which satisfy following ...
0
votes
0answers
18 views

Question about a particular estimate in Riemannian geometry.

I have been studying the book Some Nonlinear Problems In Riemannian Geometry - Thierry Aubin. On page $46$ he begins the proof of the Sobolev imbedding theorem to manifolds. The proof is divided in ...
0
votes
0answers
31 views

Integration: Countable Additive Measure?

When considering Bochner's theory of integration one notices that having a countable additive measure rather than merely additive measure is not important, or do I miss something?
1
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0answers
22 views

Numerical integration of $\sin(p_{m})$ and $\cos(p_{m})$ for a polynomial $p_{m}$

I was wondering if anyone knew about any numerical methods specifically designed for integrating functions of the form $\sin(p_{m})$ and $\cos(p_{m})$ where $p_{m}$ is a polynomial of degree $m$. I ...
3
votes
0answers
36 views

2D Integral of Bessel Function and Gaussians

I've run into the following integral, and I'm not sure how to evaluate it. $$F(k)=\int ...
0
votes
1answer
38 views

Resources for learning integral calculations

I am willing to learn about integrals . So i wonder is there any systematic book about the topic that goes progressively in difficulty and complexity . My current level is about knowing the basic ...
1
vote
1answer
41 views

Haar measure on $G \times G$, where $G$ is compact

Let $G$ be a compact group. Let $\mu'$ and $\mu$ be the Haar measure on $G \times G$ and $G$, respectively, and further such that $\mu'(G \times G) = 1$ and $\mu(G)=1$. Does it follow that $\mu' = \mu ...
0
votes
1answer
8 views

How to calculate the length of a cubic hermite spline between two points

I am using the following equation to create a cubic hermite spline: $$p_n(t) = a_nt^3+b_nt^2+c_nt+d_n$$ $$1\geq t\geq 0$$ $p_n(t)$ is the unit interval interpolation equation for dimension n. $t$ is ...
0
votes
1answer
41 views

Calculus, find the area between two given functions

I wonder why my answers were wrong, I equaled the two functions and set them equal to zero. then I found the integral and substitute with the the given points. ex: $cosx - e^x$ integration of ...
0
votes
0answers
49 views

Exponential integral with $x^2$ and $\cos x$

The first part is just a Gaussian integral and the second is the modified Bessel function of the first kind for $n=0$, but I couldn't find any information and what to do with their summation. Any tips ...
0
votes
0answers
24 views

Basic facts related to Haar measure

I have a compact group $G$ and continuous functions $f_1, f_2$ from $G$ to $\mathbb{C}$ and $g: \mathbb{R} \rightarrow \mathbb{C}$. I have two questions related to Haar meausure. Is it true that $$ ...
0
votes
0answers
8 views

Order of Romberg's method

We call a method(numerical integration) of $n-$th order, if it can integrate any polynomial of degree $n-1$ without any error. In this sense: The simpson rule is of $4$-th order and the trapezium ...
6
votes
5answers
731 views

What's the purpose of this formula?

Just found this image on the web: Can anyone explain what's the meaning (if any) of this formula? (I did a Google image search but found no answer)
0
votes
1answer
7 views

Change of variables from intinite to bounded support.

I may be missing something simple, but I am stuck. My question: I am solving a system of partial differential equations numerically, but one of the variables can take on any value, ie $x \in ...
4
votes
6answers
90 views

If $\lim\limits_{x \to \infty} f(x) = 1$, can we have function $f(x)$, such that $\int_0^{\infty}f(x)dx$ converges

I know the Initiative answer, can anyone give a neat answer based on solid reasoning EDIT : $f(x)$ is continuous
0
votes
4answers
125 views
-1
votes
1answer
78 views

How to integrate $\int \frac{x^2+\sin x}{2x+\cos x}dx=?$

I would appreciate if somebody could help me with the following problem: Q: How to integrate $$\int \frac{x^2+\sin x}{2x+\cos x}dx=?$$
0
votes
0answers
27 views

integration to the concept of work [on hold]

A cable 50 feet in length and weighing 4 pounds per foot hangs from a windlass. Calculate the work done in winding up 25 ft of the cable.neglect all forces except gravity.
4
votes
2answers
38 views

p-norm of a function

Let $f\in L^1(\mu)\cap L^\infty(\mu)$. I have proved for any $1<p<\infty$, $f\in L^p(\mu)$, $w(p)=||f||_p$ is continuous w.r.t. $p$, and $\lim_{p\to \infty}||f||_p=||f||_\infty$. Is $w(p)$ ...
1
vote
0answers
17 views

How to compute cumulative intensity process integral?

I am faced with a basic question about counting process and its intensity process used in survival analysis. It is actually the textbook example from Aalen's Survival and Event history analysis book. ...
4
votes
1answer
99 views

How to evaluate integral $\int_{0}^{\infty} \left(\frac{1-e^{-x}}{x}\right)^n dx$.

First, according to \begin{align*} \int_{0}^{\infty} x^{-m}(1-e^{-x})^{n} \, dx =\frac{n}{1-m}\int_{0}^{\infty} x^{1-m}(1-e^{-x})^{n} \, dx -\frac{n}{1-m}\int_{0}^{\infty} x^{1-m}(1-e^{-x})^{n-1} \, ...
0
votes
2answers
17 views

Line integrals; How to set $t$ boundary?

I'm having a hard time understanding how to set t boundaries in line integrals... The question is: find the line integral of $f(x,y,z)$ over the straight line segment from $(1,2,3)$ to $(0,-1,1)$. I ...
0
votes
0answers
46 views

What is the difference between a line integral with respect to x or y and a Riemann integral with respect to x or y?

I'm finding the concept of line integrals with differentials including dx or dy hard to swallow intuitively. Specifically, I'm having trouble differentiating them from a Riemann integral. What are the ...
0
votes
0answers
18 views

integration coordinates

Could anyone give me hint on how to do it? I know that I have to find the y values by: F(b)= F(a) + a-b integral f(x) dx F(b) = 150 + a-b integral f(x) dx but how to find the integral from 0 to ...
0
votes
1answer
21 views

Double integral via Riemann sum

How do I integrate the function $f(x,y)=15(x^{2}+y^{2})$, in $Q=[0,1]\times[0,1]$ via Riemann sum? I tried to get the partition $$0=x_{0}<x_{1}<\ldots<x_{n}=1\quad\text{and}\quad ...
3
votes
2answers
69 views

How to integrate $\int_{-\infty} ^\infty \frac{\cos(xy)}{x^2+1}dx$

Is there a standard trick to compute this integral for $y\ge 0$? $\int_{-\infty} ^\infty \frac{\cos(xy)}{x^2+1}dx = \int_{-\infty}^{\infty}\frac{y \cos(x)}{x^2+y^2}$ Hopefully the same trick could ...
1
vote
3answers
29 views

integral of the sphere describing lambertian reflectance

A Lambertian surface reflects or emits radiation proportional to the cosine of the angle subtended between the exiting angle and the normal to that surface. The integral of surface of the hemisphere ...
0
votes
0answers
15 views

What is correlation kernel and compare with gaussian kernel

I read a paper that said about correlation kernel that defined: $$W(x-y)=(α/1+d(|y − x|))$$ where $α =  (\int(1+d(y − x)dy)^{-1}$, $(d(|y − x|))$ is spatial Euclidean distance from the central ...
2
votes
1answer
68 views

Calculate the areas in a circle

Short: I want to calculate the areas drawn in this picture: The coordinates P00, P10, P01, P11 and Pdata are given Long: I am a programmer and want to calculate these areas, but unfortunately I am ...
2
votes
0answers
60 views

${\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$ and $\int_{0}^{\pi/2} \frac{\log \cos x}{x^2}\:\mathrm{d}x$

I have found the following new result connecting to rational log-cosine integrals. Proposition. \begin{align} \displaystyle & {\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} ...
4
votes
3answers
150 views

Definite integral $\int_0^{2\pi}(\cos^2(x)+a^2)^{-1}dx$

How do I prove the following? $$ I(a)=\int_0^{2\pi} \frac{\mathrm{d}x}{\cos^2(x)+a^2}=\frac{2\pi}{a\sqrt{a^2+1}}$$
-1
votes
0answers
43 views

Evaluating $\int\frac{1}{(x^2-5)^{0.5}}\,d(x^2+5).$ [on hold]

How can I evaluate $$\int\frac{1}{(x^2-5)^{0.5}}\,d(x^2+5)?$$ Thanks in advance!
0
votes
1answer
43 views

How can the signed area be 0?

How can the signed area be 0? For example if you have 3 on positive x side and 3 on the negative x side then you get the signed area of 0? How can area be 0?
0
votes
0answers
18 views

Normalizing a probability density function

I need to find a normalization term $N(\alpha,\beta)$ for the probability density function: $$PDF(\alpha,\beta)=(x-x_1)^{\alpha}e^{-\beta(x-x_1)}$$ In other words, solve the following equation: ...
0
votes
1answer
21 views

Unique solution for $\int_x^1 f(t) dt = 2x$ and $|x| < \epsilon$

Let $f$ be continuous on $\mathbb{R}$ such that $$f(0) \neq -2 \quad\text{ and } \quad \int_0^1 f(t) = 0.$$ Show that there exists $\epsilon > 0$ such that the equation $$\int_x^1 f(t) dt = 2x$$ ...
1
vote
1answer
22 views

Line integrals and parametrization

I've just learned about line integrals, and I need some help understanding an example problem in my textbook. The question is supposed to be really easy. Integrate $f(x,y,z)=x-3y+z$ over the line ...
2
votes
2answers
72 views

How find the function $f(x)$ such $\int_{0}^{\pi}f(x)\cos{(nx)}dx=0$

let $f(x)$ is Continuous function on $[0,\pi]$,and for infinite positive integer $n$ such $$\int_{0}^{\pi}f(x)\cos{(nx)}dx=0$$ Find the $f(x)$? I think the answer is $f(x)=c$?,But maybe have ...
1
vote
0answers
58 views

Can there be a power series with interval of convergence $[k, \infty)$?

My answer : NO Because Interval of convergence is of the form $(a-R, a+R)$ Where $a$ is centre of convergence. If there exists a power series with Interval of convergence $[k, \infty)$ $ $ We ...
0
votes
0answers
29 views

Asymptotic analysis if t tends to infinity [on hold]

Asymptotic analysis if t is large. p=1 is making contribution to the asymptotic behavior?
2
votes
3answers
82 views

Find $x > 0$ for which $\int_{0}^{x} [t]^2 \ dt = 2 (x-1)$.

What are all possible $x > 0$ for which the following equation is satisfied? $$\int_{0}^{x} [t]^2 \ dt = 2 (x-1),$$ where $[.]$ denotes the bracket (or floor) function. I guess we will have to ...
0
votes
1answer
35 views

pdf and cdf of a product of two random variables

I have a question for my probability class that I was struggling with. I found an answer online but I don't really like this answer. The question reads: Let $X$ and $Y$ have the pdf $f(x,y)= 1$ ...
1
vote
1answer
56 views

Double integral over complicated region

Suppose we wanted to compute $\iint\frac {1}{1 + x^2 + y^2} dxdy$ over the region $\frac {(x-1)^2}4 + \frac {(y+2)^2}9 \leqslant 1$. It gets quite hairy if we use elliptical polar coordinates i.e. ...
1
vote
1answer
44 views

How to solve this seemingly simple triple integral?

$$\iiint_D x^2+y^2+z^2\,dxdydz$$ $D$ is bound by $x=0, y=0,z=0$ and $x+y+z=a$, calculated by rote, I got $\frac{a^5}{20}$, is there any simpler way to do this? I tried using spherical coordinates, but ...
5
votes
2answers
132 views

Prove that $f$ is constant on $[a,b]$

$\displaystyle \int_{a}^{b} f^2(x) \, \mathrm{d}x$ = $\displaystyle \int_{a}^{b} f^4(x) \, \mathrm{d}x$ = $\displaystyle \int_{a}^{b} f^3(x) \, \mathrm{d}x$ And $f$ is continious on $[a,b]$ and ...
0
votes
1answer
92 views

How does one graph $\sum_{x=0}^{n}$ [on hold]

How does one graph a summation, like $$\sum_{x=0}^{n} n$$ Can it be like this Because if you take the points from the summation (0,0), (1,1), (2,3), (3,6) you can tell by summations it only works ...
1
vote
0answers
31 views

Error bound by the Simpson's rule

My lecture notes have a little exercise. Two functions are given: $$ f(x) = \cos(x) \ \text{and} \ g(x)=\sqrt{x+1} $$ And we're asked about the error bound of the Simpson's rule to estimate the ...
0
votes
0answers
29 views

Sketching the graph of a function with three real roots

I need to solve the following question: Sketch a graph of a function $f(x)$, continuous in all $x \in \Bbb R$, knowing that $f$ has three real roots, that $\lim_{x\to+\infty} \left[f(x)-\frac ...
0
votes
1answer
30 views

Volume of a solid in R3

How can I find the volume of this field? : $$ G=\{\left. (x,y,z) \, \right| \, x^2+y^2+z^2 \le 16 \wedge 1 \le x+y+z \le 2\}. $$ Can anybody help me? Thanks.
2
votes
2answers
63 views

When not to use integration by parts?

I am trying to evaluate this integral using integration by parts. $$I=\int_{0}^{\infty}f(x)g'(x)dx,$$ where $f(x)=\sin x$ and $g'(x)=\dfrac{x}{1+x^2}$. So: $f'(x)=\cos x$ and ...