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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-1
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2answers
43 views

Antiderivative of $xe^{-cx^2}$

I need to define $c$ in $$\int_0^\infty xe^{-cx^2},$$ so that it becomes a probability-mass function s(o that it equals 1). Where do I even begin finding the antiderivative of this? I know the answer ...
0
votes
0answers
14 views

nonnegative Riemann-integrable function, infimum

$f$ is a nonnegative Riemann-integrable function on $(0,1)$ and $f(x)\ge\sqrt{\int_0^xf(t)dt}$ for $x\in(0,1)$. Find $\inf\frac{f(x)}{x}$ I have no idea how to work out the assumption, for equality ...
1
vote
2answers
51 views

Area between a semicircle and a 45° line

I'm studying for a Calculus test and I met the following question: There's a semicircle $$y=\sqrt{1-x^2}$$ and a line at 45° degrees v=x. The task was to find the area in the positive quadrant. I ...
2
votes
0answers
7 views

When can we move a Fréchet derivative under a Lebesgue integral?

Under what conditions can we move a Fréchet derivative under a Lebesgue integral? Specifically, when does $$ G'(x) = h\in X\mapsto \int_{\Omega} \left(F_x^\prime(x,t)h\right) \mu(dt) $$ where $$ ...
0
votes
0answers
16 views

follow-up question to Hake's theorem in Bartle's book

My question is based in here. Why is it that $b$ forces to be a tag of $[x_{m-1},b]$? I can't get the right trick. Can you please give me some hints? Thanks
2
votes
1answer
66 views

How to calculate the integral of $f(x)$? [on hold]

Let $f(x)$ be a function which satisfies the following two properties: 1) $f(x) + f(-x) =2$ 2) $f(1-x) = f(1+x)$ I need to calculate the $\int_0^{2016} f(x)dx$. I already tried to find $f(x)$ ...
2
votes
3answers
75 views

Calculate an integral depending on n

Is there a way (simple or not) to calculate the following integral? $$\int_{-1}^{1} \sqrt[n]{1-x^n} dx$$ Thanks
3
votes
0answers
26 views

Definite integral of arcsine over square-root of quadratic

For $a,b\in\mathbb{R}^{+}\land0<a<1$, define $\mathcal{I}{\left(a,b\right)}$ by the integral ...
1
vote
1answer
25 views

Integral with an unknown function

I am trying to solve this integral $$ \int \frac{f(x)}{g(x)}\frac{\mathrm dg}{\mathrm dx}\mathrm dx $$ where $g$ is an unknown function of $x$, and $f(x)$ is a known function that can be integrated ...
2
votes
1answer
26 views

Radon-Nikodem Derivative of a purely nonatomic Borel Measure

If $\mu$ is a purely non-atomic Borel measure on a topological space $X$ then must its density be a continous function to $\mathbb{R}$? My intuition says yes because all my counterexamples are not ...
1
vote
0answers
33 views

Weird question about interval of convergence

The question is: if $$ f(x) = \sum \limits_{n=0}^\infty x^n$$ determine the interval of convergence for the power series representation of $$\int_0^x f(t) \, dt$$ That integral threw me off.
1
vote
1answer
40 views

Spivak Calculus Ch. 19 #15

(a) Find $\int \sin^4 x\, dx$ in two different ways: first using the reduction formula and then using the formula for $\sin^2x$. (b) Combine your answers to obtain an impressive ...
-3
votes
0answers
14 views

How to find volume using Riemann sum with expression $e^x + 3x^3 - x^2$? [on hold]

I'm desperate. It is for math class so please help if you know how to find volume using Riemann sum, and not double integral.
0
votes
1answer
26 views

Parametric integral and equivalence in $\infty$

I have to find a equivalent when $x$ comes to $\infty$ for all $a$ (fixed) in $\mathbb{R}_+^*$ of this integral : $$ \int_0^a \frac{e^{-xt}}{\sqrt{a-t}}\mathrm{d}t $$ My work : For $x \in ...
0
votes
0answers
19 views

$\int_{0}^{+\infty}\frac{x^2+x^3}{x^2\cdot \left | x-1 \right | \cdot \frac{3}{4} \left |x-4 \right |^{\frac{4}{3}}} dx$ convergence

Does $$\int_{0}^{+\infty}\frac{x^2+x^3}{x^2\cdot \left | x-1 \right | \cdot \frac{3}{4} \left |x-4 \right |^{\frac{4}{3}}} dx$$ converge? Domain of this integrand is $x \in \mathbb{R} : x\neq 0, ...
3
votes
3answers
217 views

double integral $\int_0^t \int_0^s \frac{\min(u,v)}{uv} \, dv \, du$

I want to calculate the double integral: $$\int_0^t \int_0^s \frac{\min(u,v)}{uv} \, dv \, du$$ I don't know how to o that even if it seems simple. Thanks in advance for your help
1
vote
1answer
12 views

Sequence from generating function with integral

So, let $A(x)$ be the generating function of $a_0,a_1,\dots$ then what would be the sequence of the generating function: $$\int^x_0 A(t)dt$$ Since I am not much acquainted with integrals any help ...
1
vote
0answers
17 views

Infinite sums over integral of triple associated Legendre polynomials

I have a couple of integrals of triple infinite sums of associated Legendre polynomials, which I'd like to integrate using Gaunt's Formula. Any help would be very much appreciated, as I'm really ...
0
votes
0answers
26 views

Continuous convergence [on hold]

If f_n converge pointwise to $0$ in $\mathbb{R}^d$, $\int f_n dm =1$ for every $n\in \mathbb{N}$ and $g \in L^1_m \cap C(\mathbb{R}^d,\mathbb{R})$. Then how can I prove that: \begin{equation} \int ...
1
vote
1answer
29 views

Area Between Intersecting Lines - Elegant Solution?

I am running simulations, and the output will be a line y = mx+b. I am interested in the area below the line between x=0 and x=1. I am only interested in the area that is below the diagonal y = x. I ...
2
votes
1answer
56 views

integrate this double integral by any method you can. [on hold]

I'm having trouble with this double integral: $$\int_0^2\int_0^{2-x} \exp\left(\frac{x−y}{x+y}\right)\text dy\,\text dx$$
0
votes
0answers
16 views

Strongly continuous semigroup Kolmogorov forward integral equation

Let $\{ P_t \}_{t \geq 0}$ be a SCSF($\mathcal{S}$) (strongly continuous semigroup on $\mathcal{S}$) on the space $(E,\mathcal{E})$, where $E$ is a Polish space, equipped with the ...
3
votes
1answer
59 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
3answers
79 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} ...
3
votes
2answers
278 views

Calculating the integral $\int_{1/3}^{3}\frac{\arctan(x)}{x^2-x+1}\;\mathrm{d}x$

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
vote
3answers
35 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
53 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 ...
0
votes
1answer
34 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
vote
0answers
26 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 ...
0
votes
0answers
16 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
17 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
22 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
240 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
vote
1answer
31 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
2answers
71 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 ...
3
votes
4answers
103 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
vote
1answer
48 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
63 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
votes
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
vote
1answer
27 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
27 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 ...
0
votes
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, $$ ...
-2
votes
2answers
54 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
91 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
65 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
votes
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
358 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 ...