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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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 ...
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1answer
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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 ...
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1answer
5 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 ...
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1answer
37 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 ...
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0answers
44 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 ...
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0answers
23 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 $$ ...
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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 ...
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5answers
573 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)
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1answer
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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
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6answers
88 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
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4answers
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1answer
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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 $$\int \frac{x^2+\sin x}{2x+\cos x}dx=?$$
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0answers
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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.
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2answers
37 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 $w(p)$ ...
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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
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1answer
97 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} \, ...
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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 ...
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0answers
45 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 ...
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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
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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
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2answers
66 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 ...
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3answers
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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 ...
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0answers
14 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 ...
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0answers
54 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
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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
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3answers
146 views

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

How do I prove the following? $$ I(a)=\int_0^{2\pi} \frac{\mathrm{d}x}{\cos^2(x)+a^2}=\frac{2\pi}{a\sqrt{a^2+1}}$$
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0answers
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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?
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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: ...
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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 $$\int_x^1 f(t) dt = 2x$$ ...
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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 ...
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0answers
57 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 ...
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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
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3answers
81 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 ...
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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
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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
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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
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2answers
129 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 ...
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0answers
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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 ...
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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 ...
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1answer
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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
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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 ...
0
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1answer
31 views

Constructing the graph of a function

I need to solve the following problem: Consider the function $g(x) = \ln(x^{2}) + 2$. Construct the functions graph $f(x)=\int g(x)\:\mathrm{d}x$ considering the integration constant equal to ...
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0answers
46 views

Solution to the integral?

What is the solution of the following integral: $$ \int_{-1}^{K} x^{B+1} e^{-Nx} dx $$ where $N$ and $B$ are constants
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For which $a>0$ does this Lebesgee-integral exist (and is finite) [on hold]

Let $\lambda$ be the Lebesgue-measure over $(\mathbb{R},\mathbb{B})$. Determine, for which $a>0$ the Lebesgue-integral: $$\displaystyle\int_\pi^\infty \left(\frac{\sin x}{x}\right)^{a}\text{ ...
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0answers
29 views

$ \int_0^\infty (1+t^2)^{-s} (1+it)^{s'} 2t \; d t.$

The following integral bothers me since weeks: $$ \int_0^\infty (1+t^2)^{-s} (1+it)^{s'} 2t \; d t.$$ Has any body a suggestion for this integral. $Re\; s >0$ sufficiently large and $s'$ an ...
1
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1answer
30 views

consider a square of side length $x$, find the area of the region which contains the points which are closer to its centre than the sides.

Any ideas how to start. I am having trouble figuring out the region itself All ideas are appreciated thanks
1
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1answer
34 views

Moving $d$ terms inside a double integral?

I was dealing with an integral expression like that: $$\int zf(z)dz$$ In this term it is known that $f(z)=\int g(x,z) dx$. So I can replace $f(z)$ in the first term like that: $$\int z(\int ...