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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0answers
25 views

Integral does not 'converge' despite describing a well-defined area…

I have almost evaluated (where all variables are real including the variable $i$) $$ C_1\int_{a + bt^2}^{i} \frac{r ...
0
votes
2answers
50 views

How do I calculate $ \int_{1}^{3} x/(2-x) \;\mathrm{d}x$

$ \int_{1}^{3} \frac{x}{2-x} \;\mathrm{d}x$ $ \int_{1}^{2} \frac{x}{2-x} \;\mathrm{d}x$ + $ \int_{2}^{3} \frac{x}{2-x} \;\mathrm{d}x$ $u = 2-x$ $\lim_{e\to0} \left[ \int_{-e}^{1} \frac{2-u}{u} ...
2
votes
2answers
86 views

Calculating an integral

Can somebody help me calculate the following integral: $$\int\limits_{1/3}^{3}\frac{\arctan(x)}{x^2-x+1}\;\mathrm{d}x$$ I have tried integration by parts, but I got stuck in it. Wolfram also didn't ...
1
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3answers
28 views

Parametrization of $x^2+y^2-ay=0$

I am to find the circulation of $$y^2 dx + x^2 dy$$ along the (counterclockwise) path $$\Gamma : x^2+y^2-ay = 0$$ both with and without using Green's theorem. Apparently, $\Gamma$ is supposed to ...
2
votes
3answers
41 views

Which function can be used for Substitution

Find the value of $$I=\int_{0}^{\frac{\pi}{2}}\left(\sin(x)-\cos(x)\right)\,\log(\sin(x))dx$$ Method $(1)$. I splitted up the Integral into two Integrals as $$I=I_1+I_2$$ where ...
1
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0answers
33 views

proving a statement on Measure theory

Consider $(\Omega, U, \mu)$ be a measure space and X be an integrable function and for $A$, $\{A_n\}\in \mathscr{U};n\in \Bbb N$ I need to show that $\int_{A_n}X d\mu \to_{n\to \infty}\int_A Xd\mu$ ...
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1answer
28 views

Example of a convergent series for which integral test fails?

Is there example of a convergent series for which integral test fails or can not be applied? Just wondering if integral test is the silver bullet of convergence tests, or are there any series that any ...
1
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0answers
22 views

Bound on the integral of a function with multiple zeros

This is a follow-up to this If $f(0)=f(1)=f(2)=0$, $\forall x, \exists c, f(x)=\frac{1}{6}x(x-1)(x-2)f'''(c)$ Let $f:[0,2]\to \mathbb R$ be a $C^3$ function such that ...
-2
votes
1answer
52 views

Calculate this double integral [on hold]

Recently took and exam and this was one of the questions and I wanted to check if I did it right Let $R$ be the triangular region in the ($x$,$y$)-plane with vertices $(0,0)$, $(1,0)$ and $(1,2)$. ...
0
votes
1answer
34 views

Integral with 'reset'

I am trying to mathify the following algorithm description: The algorithm iterates over the elements in the sequence $(f_1, ..., f_n)$, calculating the heuristic function $h(f_k, f_{k+1})$ for ...
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0answers
10 views

Definite integral involving Legendre Polynomial

Does anyone know the answer to the following definite integral: $\displaystyle \int_{0}^{\pi}P_{\ell}(\cos\theta)\sin^{k}\theta\, d\theta$ for $k\geq1$, where $P_{\ell}(x)$ is the $\ell$-th Legendre ...
0
votes
0answers
14 views

integrating non functions

Does it make sense to integrate non functions for example what does it mean to integrate $ x^2 + (y-1)^2 = 1 $ I think you can integrate the above parametrically $ x= sin(t) $ $ y=cos (t) + 1 $ but ...
0
votes
0answers
21 views

Integrating $\operatorname{Log}(z+2)$ along the unit circle [duplicate]

For the function $f(z) = \operatorname{Log}(z + 2)$, where we choose the principal branch of logarithm (namely, $−\pi < \operatorname{Arg}(z) < \pi$), and the contour $C := \{z \in ...
2
votes
4answers
225 views

Cauchy integral formula

Can someone please help me answer this question as I cannot seem to get to the answer. Please note that the Cauchy integral formula must be used in order to solve it. Many thanks in advance! ...
1
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1answer
28 views

Integration — Work done in pulling an elevator using a rope [on hold]

An elevator weighing 3000 lbs. is supported by a 12 ft. cable that weights 14 lbs./ft. Find the work a which has to do by pulling the rope to lift the elevator 9 feet. I eventually figured out ...
2
votes
1answer
48 views

Integrating a square's perimeter to get its area

I am trying to wrap my head around some integration applications. I went through the exercise of integrating the circumference of a circle, $2*\pi*r$, to get the area of a circle. I simply used the ...
2
votes
4answers
97 views

Why the anti derivative of $\sec(x) \cdot \tan(x)$ is $\sec(x)$?

I have discovered that $$\sec(x) = \frac{1}{\cos(x)}$$ but I do not understand why the indefinite integral of $\sec(x) \cdot \tan(x)$ is $\sec(x)$. I am watching the following videos: ...
1
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1answer
46 views

A real integral (may be requires contour integration)?

The integral I have in mind is $$\int^\infty_0 x^{r}(x + \lambda)^{-1}dx$$ where $r \in (-1, 0)$, and $\lambda$ is a non-negative constant. I apologize if this is really easy and I am missing some ...
2
votes
0answers
57 views

Geometric proof for Sophomore's dream

Is there a "visual proof" for sophomore's dream? $$\int_0^1 x^{-x}\,dx = \sum_{n=1}^\infty n^{-n}.$$ In the wikipedia article there are two algebraic proofs, but the integral and the summation has ...
0
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1answer
15 views

use laplace transform to solve the given integral equation

use Laplace transform to solve the given integral equation I don't know how start because it differences on other Laplace question I see before
1
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1answer
26 views

curves and integral

Find the area between these curves. $$y=\dfrac{3}{2x+1},\qquad y=3x-2;\qquad x=2\quad \text{et} \quad y=0 $$ indeed, I calculate the integral of the blue function between $1$ and $2$. Then, I ...
1
vote
0answers
23 views

Integral involving Whittaker function

Consider the following integral: $$ \int_1^{\infty} \frac{e^{u/2}}{u}[-\mathrm{Ei}(-u)]\,W_{1,\imath p}(u)\,du, $$ where $\imath=\sqrt{-1}$ and $p>0$ selected so that $W_{1,\imath p}(1)=0$; here ...
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0answers
18 views

Mixture of binomial distributions

I have a population of agents with a single real-valued attribute $x$. Each of them performs $n$ Bernoulli trials with success probability $q(x)$ which depends on their attribute. In particular, $$ ...
-1
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2answers
44 views

A question related to measura space

Let a real value $X$ be a random variable and consider $\int_{\Omega}|X|dP \lt \infty $. I need to show that \begin{equation*} nP(|X|\gt n)\to_{n\to \infty} 0. \end{equation*} please help me ...
2
votes
1answer
88 views

Study the following integral: $\int_0^\infty \frac{\mathrm{d} x}{x \cdot \ln x \cdot \ln^{(2)} x \cdot \ln^{(3)} x … (\ln^{(k)} x)^s }$

How do I calculate for which values of $s$ the following integral converges? $$\int\limits_{0}^{\infty} \frac{\mathrm{d} x}{x \cdot \ln x \cdot \ln^{(2)} x \cdot \ln^{(3)} x \cdots (\ln^{(k)} ...
2
votes
1answer
64 views

Triple integration, sphere (electric field).

The sphere $K\subseteq R^3$ with radius $R$ has a homogeneous charge density $\rho$. Find the electric field E, produced by K outside of, meaning, find the integral ...
0
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0answers
17 views

The area of a stereographic projection

I'm newbie at multidimensional integration and I'm trying to make a working algorithm in Wolfram that can help me compute the areas on the unit sphere without complicated parametrization, provided I ...
1
vote
0answers
35 views

Calculating total mass of a wire

I'm giving the following $$ \delta(x) = x + 7,\quad (0 \leq x \leq 4) $$ It says you are given the length-density function, $\delta(x)$, of an ininfinitesimally thin wire lying on the $x$-axis over ...
3
votes
4answers
350 views

How to solve certain types of integrals

I'm asking for a walk through of integrals in the form: $$\int \frac{a(x)}{b(x)}\,dx$$ where both $a(x)$ and $b(x)$ are polynomials in their lowest terms. For instance $$\int ...
2
votes
1answer
40 views

Evaluating a triple integral by inspection

I would like to evaluate the triple integral: $$\iiint\limits_D {2 + 3{x^2} + 3{y^2}dV}$$ where $D$ is a conic domain with vertex $(0,0,b)$ and axis along the $z$-axis with a base (disk) with radius ...
0
votes
1answer
24 views

Finding the complex Fourier series… [on hold]

Here's is the problem: And this is my solution for part (i) of the problem. Is it correct? Can I simplify it more than I did? For part (ii), I don't know how to start. So please give me a ...
-4
votes
1answer
49 views

How can I calculate this integral? [on hold]

Calculate the integral: \begin{equation*} \lim_{n\rightarrow\infty}\int_{0}^{2\pi} \frac{\ cos\ nt} {\ cos\ (t)+4} dt. \end{equation*}
5
votes
2answers
123 views

Infinite integrals$\int_0^{ + \infty } {\frac{1}{{\left( {x + 1} \right)\left( {{x^n} + 1} \right)}}dx} .$

How to calculate $$\int_0^{ + \infty } {\frac{1}{{\left( {x + 1} \right)\left( {{x^n} + 1} \right)}}dx} .$$
2
votes
2answers
24 views

Cylinder cut out of a sphere. (volume).

1.) A cylinder $Z=${$(x,y,z)\in R^3|x^2+y^2\leq \rho^2$}$~~$ ($0<\rho<R$) is cut out of a sphere $K\in R^3$ with radius $R>0$ centered around the origin. Find the volume of the rest ...
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votes
1answer
40 views

Integration by parts prove integral of cos^n x dx [on hold]

I'm having a problem with one of my questions. How can I prove that $\begin{align}\int\cos^n x dx&=\sin x\cdot\cos^{n-1}x+(n-1)\int\sin^2x\cos^{n-2}x dx\end{align}$ ?
4
votes
1answer
25 views

Double integral (and for the area enclosed by a lemniscate).

a) Transform into polar coordinates and compute the integral $\int_\Omega ln(1+x^2+y^2)d(x,y)$ where $\Omega$ is the interior of the unit circle in the first quadrant. b) $\Omega\subseteq R^2$ ...
2
votes
1answer
60 views

What can the various ways of integrating $\int \frac { x ^2 }{ (x \sin (x) +\cos(x))^2} \, dx$

$$ \int \frac {x^2}{(x\sin(x) + \cos(x))^2} \, \mathrm{d}x $$ Well I found a method for solving this sum in a book saying that : We can multiply and divide the expression by $x\cos(x)$ and then ...
0
votes
0answers
27 views

Can this definite integral of an inverse Laplace transform by simplified?

Can either of the below expressions involving an unknown analytic function $h(s,t)$ and the inverse Laplace transform $\mathcal{L}^{-1}$ be simplified? $$ \int\limits_{0}^1 \mathcal{L}^{-1} \left\{ ...
0
votes
3answers
52 views

integration $\int_{0}^{1/8} \frac{4}{\sqrt{(1-4x^2)}} \,dx$

integration equation $$\int_{0}^{1/8} \frac{4}{\sqrt{(1-4x^2)}} \,dx$$ my work $t= \sqrt{(1-4x^2)} $ $dt = -4x/\sqrt{(1-4x^2)} dx $ stuck here also
0
votes
2answers
41 views

I need help with the integration order please

the integral is as follows: find the volume between these regions bounded by : $z = x^2 + 3y^2$ and $z = 9 - x^2$ I discovered that this would be the space bounded by the elliptic paraboloid and the ...
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votes
2answers
35 views

solve $(x^2 y+y)dy/dx =\arctan(x)$ , $y(1) = -1$

solve the equation $(x^2 y+y)(dy/dx) = \arctan(x)$ , $y(1) = -1$ my work until now $(x^2 y+y)\,dy = \arctan(x)\,dx$ $x^2 y\,dy+y\,dy = \arctan(x)\,dx$ ---> stuck here
2
votes
3answers
40 views

Integral Question using the Rule of Subsitution

I'm confused as to why $ \int e^{kx}dx$ = $\frac{e^{kx}}{k} + C$. I'm using the rule of substitution and came to the conclusion that it should be $e^{kx}k$ because the derivative of $kx$ is $k$. What ...
0
votes
0answers
27 views

Gradient and Laplacian in integral.

Let $u,v,f$ be functions of $\mathbb{R}^n$ to $\mathbb{R}$, with compact support in a domain $U$, this formula $$\int_{U} f(x) (Du \cdot Dv) dx = \int_{U} f(x)(u D(Dv)) dx = \int_{U} f(x) u(x) \Delta ...
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0answers
45 views

Calculate integral

I have to calculate $$\operatorname{PV}\int_{-\infty}^{\infty}\frac{1}{\pi}\frac{y}{1+y^2}dy$$ I ended up with ...
0
votes
0answers
17 views

Using polar coordinates in this integral

I'm trying to solve something along the lines of: $$\iint \frac{\partial F_1(x,y)}{\partial x}+\frac{\partial F_2(x,y)}{\partial y}dydx$$ which I want to change to polar coordinates, but I don't ...
2
votes
4answers
58 views

integration $\int_{0}^{3}(12/(x^2 -6x+12))\,dx$

$$\int_{0}^{3} \frac{12}{x^2 - 6x + 12} \,dx$$ I assume that $x^2 - 6x + 12 = (x-3)^2 + 3$, then $t = x - 3 \rightarrow dt = dx$ since $$\int_{0}^{3} \frac{12}{t^2 + 3}\,dt$$ and now I am stuck. ...
1
vote
1answer
79 views

Feynman Integration Problem

$$ I = \frac{\pi^2}{8} - \int_0^1 \frac{\arctan(x)}{\sqrt{1-x^2}} \,dx $$ Evaluate $I$ $$ I = \frac{\pi^2}{8} - \int_0^1 \frac{\arctan(x)}{\sqrt{1-x^2}} \,dx$$ $$f(a) = \int_0^1 ...
3
votes
2answers
62 views

$\int \frac{\exp (z)(\sin(3z)}{(z^2-2)(z^2)} dz$ on $|z|=1$

So I need to calculate \begin{equation*} \int \frac{\exp(z) \sin(3z)}{(z^2-2)z^2} \, dz~\text{on}~|z|=1. \end{equation*} So I have found the singularities and residues and observed that the ...
0
votes
0answers
24 views

Can limits to definite integral be vectors?

There are two cases: 1-Limits are scalar and function to be integrated is vector. 2- Limits are vector and function to be integrated is vector. Are both valid. If yes can you give example for each. ...
2
votes
1answer
24 views

Calculating volume by shell integration

$y = \ln x$, region is delimited by $y = -1$, $y = 2$ and the $y$-axis, it rotates around the $y$-axis. It's quite simple to solve by using disk integration but I can't get it right with shell ...