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 ...

learn more… | top users | synonyms (2)

4
votes
5answers
50 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
106 views
-1
votes
1answer
68 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
35 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
16 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
90 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
41 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
17 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
19 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
64 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
23 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
12 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
0answers
50 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
54 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
2answers
117 views

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

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
20 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 $$\int_x^1 f(t) dt = 2x$$ ...
0
votes
0answers
22 views

Radon measure, Integral

Let $d \geq1$ and $D\subset \mathbb{R}^{d}$ be open We define for $u \in C_{0}^{\infty}(D)$ \begin{eqnarray*} S(u)=\int_{D\times D \backslash {\rm diag} } |u(x)-u(y)|^{2} J(dx,dy) \end{eqnarray*} ...
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
71 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
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 ...
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
80 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
34 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
55 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
42 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
127 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
29 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 ...
0
votes
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 ...
0
votes
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
-1
votes
0answers
20 views

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{ ...
0
votes
0answers
28 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
vote
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
vote
1answer
31 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 ...
-1
votes
1answer
23 views

Integral curves and ODE [on hold]

How do I find a first-order differential equation having the given family ofcurves as integral curves to that one: all circles through the points $(1, 1)$ and $(-1,-1)$?
0
votes
2answers
9 views

Volume integral of a gradient

Assuming we have a scalar function f which tends to 0 at the boundary of a space. Is it true that the volume integral of the gradient of f will also tend to 0?
-1
votes
1answer
52 views

integral with pdf of a gaussian [on hold]

$$ I = \int_{0}^{\infty} x \phi(x) dx $$ where $\phi(x)$ is the pdf of a normal distribution. Here I read that: If $X = \mu + \sigma U$ with $U$ a std normal, $$ I = E[\mu + \sigma U; mu + \sigma ...
0
votes
3answers
51 views

Integration and ODE

How do I integrate this? $$y'=1+\frac{y}{x}$$ I just don't know how to start. I think I gotta to try some variable changing, but I don't think I'm gonna so far with this.
1
vote
1answer
41 views
-2
votes
0answers
27 views

How to solve this double integral? [on hold]

$$\int_{0}^{2\pi}\int_{0}^{2\pi} f(\omega_st,\omega_0t)\sin(\omega_st+\omega_0t)\,d{\omega_st}\,d{\omega_0t}$$ How to calculate? Thank you!
0
votes
3answers
53 views

Find $\int_0^4\int_{0}^{4}xy \sqrt{1+x^2+y^2} \,dy\, dx $

I am having a tough time figuring this one out. Any help will be appreciated. do we have to approximate, or can we actually find it
1
vote
1answer
36 views

Volumes of Revolution Washer Method

I have to find the volume of revolution of a region called $C$ using around the $y=-1$ axis. The region is bounded above by $y \ = \ \ln(x+1)$, bounded below by $y=e^{-x}$ and on the right by $x=3$. ...