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Questions tagged [integration]

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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1answer
18 views

How to find integral of the given function [duplicate]

I have a question which asks us to solve the following integral: My approach: $(1+\sin(2x)^{1/2}=[(\sin x)^2+(\cos x)^2+(2\sin x)(\cos x)]^{1/2}=[((\sin x)+(\cos x))^2]^{1/2}$ What I did in the ...
1
vote
0answers
43 views

Integral of $(1-x^a)^\frac{1}{a}$

Does the integral $$\int (1-x^a)^\frac{1}{a} dx$$ have a closed form solution? The parameter $a$ is a constant and both $a$ and $x$ are positive real numbers less than or equal to $1$. I tried ...
0
votes
0answers
28 views

Evaluation of $\int_{0}^{\infty}\frac{\sin(bx)}{x^2}\mathrm{d}x$

I'm trying to evaluate the following integral: $\int_{0}^{\infty}r^2\cdot\sin\big(\frac{b}{r^3}\big)\mathrm{d}r$ I had to solve a similar integral with cosine rather than sine and it was helpful to ...
0
votes
2answers
30 views

Study the function: $F(x)=\int_{x}^{2x}\frac{e^{2t}-e^t}{t}\,dt$

Study the function: $$F(x)=\int_{x}^{2x}\frac{e^{2t}-e^t}{t}\,dt$$ I start by saying that it is the first time I am studying a function defined by an Integral. Talking about the domain of this ...
0
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0answers
39 views

$\iint_{S} z+y+\sqrt{(a^2-x^2)} \,ds$ , surface integral

$$\iint_{S} z+y+\sqrt{(a^2-x^2)} \,ds$$ $$ S: x^2+y^2=a^2,0\leqslant z \leqslant c $$ $$ z,c>0 $$ Evaluate surface integral. I wanted to express x(y),then with it evaluate $ dS $...
0
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1answer
21 views

Multivariable integral, use of Dirac Delta and Heaviside Theta

$\newcommand{\diff}{\operatorname{d}}$ $\newcommand{\deriv}[2]{\frac{\diff #1}{\diff #2}}$ $\newcommand{\pderiv}[2]{\frac{\partial #1}{\partial #2}}$ $\renewcommand{\vec}[1]{\boldsymbol{#1}}$ I have ...
0
votes
1answer
32 views

Closed/compact form solution for $\int_{0}^\infty\int_{x_1/z}^\infty \frac{e^{-y_1(z+1)}}{1+z-\frac{x_1}{y_1}}\,dy_1\,dx_1$

I am trying to solve this double integral $$\int\limits_{x_1=0}^{\infty}\int\limits_{y_1=\frac{x_1}{z}}^{\infty} \frac{e^{-y_1(z+1)}}{1+z-\frac{x_1}{y_1}}\,dy_1\,dx_1$$ where $z> 0$. Is there ...
1
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2answers
46 views

Using fundamental theorem of calculus in Sturm-Liouville orthogonality proof

Theorem: The eigenfunctions of the Sturm Liouville BVP $$\int_0^a \sin \left( \frac{n \pi x}{a} \right) \sin \left( \frac{m \pi x}{a} \right) \ dx = 0$$ when $m \not= n$. satisfy the ...
0
votes
1answer
46 views

Proof of $\int_{-\infty}^{\infty} e^{-\pi t^{2}} e^{- i 2 \pi t v} dt = e^{- \pi v^2}$ using binominal square

I tried to proof this as a (signal engineering) homework using binomal square, but the example answer was given using differential equations. I'd like to know if my approach was possible. I tried ...
0
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1answer
31 views

Exponential integral with square root

Can this be solved? $$Y=\int_{0}^\infty Ae^{-B\sqrt{t}}\mathrm{d}t$$ Where $A$ and $B$ are constants.
0
votes
1answer
68 views

Black hole integral

What I call as ‘black hole’, has a formal name of ‘limit point of singularities’. Suppose $k$ is a black hole of the function $f$ (e.g. $0$ is the black hole of $\csc \frac1z$), then how to evaluate ...
0
votes
1answer
41 views

Length of an ellipse

I am asked to give the length of the curve: $\gamma: [0,2\pi]\to\mathbb{R}^2$, $\gamma(t)=(a\cos(t), b\sin(t))$ where $a,b\in(0,\infty)$ We get to solving the following: $\int_0^{2\pi} \sqrt{a^2\...
1
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0answers
21 views

Line integral and differentiability of $\Vert f \Vert^{n}$ where $f(x) = A(x)$

Problem. Let $f: \mathbb{R}^{n} \to \mathbb{R}$, $n \geq 2$, the function defined by $f(x) = Ax$ where $A$ is a matrix $n \times n$. For each of the statements below, show if is true or give a ...
2
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1answer
37 views

Is success realistic using u-substitution when the derivative isn't in the integrand?

If one attempts a u-substitution that leaves $x$ in the new integrand (i.e., by reintroducing an $x$ when substituting $dx$ out), it stands to reason that the $x$ could be eliminated by inverting the ...
0
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0answers
13 views

3D Gauss-Hermite Quadrature

Is it possible to examine a 3D integral by using Gauss-Hermite quadrature type technique? I mean there might be an equation like this (with analogy to 1D Gauss-Hermite quadrature): $\int_{-1}^{-1} \...
1
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4answers
78 views

Determine the limit of the integral $\lim_{n\to \infty}\int_0^1nx^nf(x)\,dx$ [on hold]

Let $f:[0,1]\to \mathbb{R}$ be a continuous function. Determine the following limit: $$\lim_{n\to \infty}\int_0^1nx^nf(x)\,dx.$$ As $f$ is continuous with closed interval so $f$ attains maximum ($M$) ...
1
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1answer
25 views

Breaking an integral $n\sum_\limits{k}^{n-1} \int_{k/n}^{(k+1)/n} \int_x^{(k+\theta)/n}f'(s) \, ds \, dx$

Let $\theta\in[0,1]$ be a constant and $f\in C^1[0,1] $. \begin{align} & S_n=\sum_k^{n-1}f\left(\frac{k+\theta} n\right)-n\int_0^1 f(x)\,dx \\[10pt] ={} & n\sum_k^{n-1} \int_{k/n}^{(k+1)/n} \...
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0answers
52 views

continuity of first derivative of integral function

following conditions are held: $1) \quad f \quad$ is integrable over $\quad[a,b]$. $2)\quad a<c<b$. $3)\quad F(x)= {\int_a ^xf(t)} d t\quad , a<x<b$. $4) \quad f\quad$ is ...
3
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1answer
55 views

Counterexample: Interchange Limit and Integral

Given continuous functions $f_n:[0,1]\to\mathbb R$ uniformly converging to 0 and $p>1$ such that $t^{-p}f_n(t)\in L_1[0,1]$, is then \begin{align*} \lim_{n\to\infty}\int_0^1\frac{f_n(t)}{t^p}dt=0 \...
3
votes
1answer
43 views

$u$-substitution failure

I'm practicing for the GRE exam, and came across the following question: If $$ f(x) = \int_x^0 \frac{\cos(xt)}{t}\, dt, $$ find $f'(x)$. The answer given is $\frac{1}{x}(1 - 2\cos(x^2))$, and I see ...
0
votes
2answers
24 views

How to interpret this part of an integration? (Areas and volumes)

This semester I've learnt how to calculate areas and volumes by using double and triple integrations, the procedure was easy and kind of easy, but there was thing that I never could really understand ...
0
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1answer
25 views

Dirac Delta in polar coordinates, integrating delta from 0 to $\infty$

Many sources mention how to recast Dirac's Delta function from Cartesian into polar or spherical coordinates. They say e.g. for polar: $$\int_{-\infty}^{\infty}\delta(\vec{x})\,{\rm d}^3x=\int_{-\...
1
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1answer
35 views

Divergence theorem and a hemisphere

I'm confused about applying the Divergence theorem to hemispheres. Here is the statement: As far as I understand, this question asks to compute $\int\int_{S_1}\mathbf F\cdot d\mathbf S$ over $$S_1=\...
0
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0answers
48 views

Recurrence involving integration

I have the following, rather methodic (than aiming at computational complexity, but still...), question. Some denotions. For a vector $\vec{x} = (x_1, \ldots, x_n)$ (of, here and thereafter, positive ...
1
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1answer
23 views

Different results after changing the order of integration with constant limits (Failure of Fubini's theorem)

I have the following question $I_{1}=\int _{0}^{1}\int _{0}^{1}\ \frac{(x-y)}{(x+y)^{3}}\ dy\,dx$ Evaulating the above I get $I_{1}=0.5$ Now if I switch the order of integration $I_{2}=\int _{0}^{...
0
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0answers
26 views

Expressing an integral in terms of the integral representation of hypergeometric function [on hold]

Can the integral, $$\int_b^x \frac{r^{15/2}}{(r^5-b^5)^{3/2}}\,dr$$ be expressed in terms of hypergeometric function? Thanks in advance.
1
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2answers
19 views

Identity of the Poisson distribution

I have recently begun to study John Kingsman's "Poisson processes", and in the first chapter, the author defines $\mathbb{P}${$X$ = $n$} = $\pi_{n}$($\mu$) = $\mu^{n}e^{-\mu}/n!$ "Differentiating ...
1
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1answer
23 views

Finding a closed form expression for an integral with parametrization

Is there a general expression if the integral below is evaluated? Note that $-\infty<q<1$ where $q$ can take either integer or non-integer values; and that $b<x$ where $b,x>0$. The ...
1
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0answers
29 views

Taking the limit of ratios of interpolation error

I don't know how to take the limit of the ratio of two functional error estimates, when both go to zero in particular $$ \lim_{h\to 0} \frac{h \| u - u_h\|^2_{L^2(\partial \kappa)} }{ \| u - u_h\|^2_{...
0
votes
0answers
44 views

function Riemann integrable but not Borel measurable

Let be $C$ the cantor set. Then there exists a set $A\subset C$ with $A\notin \mathcal{B}$, i.e. not Borel measurable. Consider $f:=\mathbb 1_A$, then f is continuous on $[0,1]\setminus C$ and $f=0 $ ...
0
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2answers
47 views

DMCT $\int fd\mu=\lim_{n\to\infty }\int f_n d\mu$ equivalent to $\lim_{n\to\infty}\int \mid f_n -f \mid d\mu=0$?

Let $f$ and $f_n$ measurable numeric functions, $n\in \mathbb N$ and $f=\lim_{n\to\infty}f_n$ a.e. and suppose an integrable $g\ge 0 $ exists with $\mid f_n \mid \le g$ a.e. Then $f$ and $f_n$ are ...
3
votes
4answers
132 views

Find the value $\int_{0}^{1}f(x) \,\mathrm dx$ without words

Let $f(x)=ax^3+bx^2+cx+d$ such that: $$f(0)=1,\quad f(0.5)=5, \quad f(1)=15$$ Find: $$\int_{0}^{1}f(x) \,\mathrm dx$$ It is said that it can be solved without words.
0
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1answer
21 views

Trouble finding inverse of CDF to sample from

I am attempting to do a discrete approximation using subintervals of a PDF's 99% CI. I understand that to generate samples from this distribution to use in my program, one can take the inverse of the ...
1
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1answer
58 views

How to compute $\int_H F\cdot n \, dS$ efficiently?

Let $H=\{(x,y,z):z>0, x^2+y^2+z^2=R^2\}$ and $$F(x,y,z)=(x^2(y^2-z^3),xzy^4+e^{-x^2}y^4+y,x^2y(y^2x^3+3)z+e^{-x^2-y^2})$$ Find $\int_H F\cdot n\,dS$ where $n$ is the outward unit normal and $dS$ ...
3
votes
1answer
76 views

Is it true that $\lim_{n\to\infty}\int_{0}^{1}f_n(x)\,dx = \int_{0}^{1}f(x)\,dx$ in general and if $|f_n(x)|\le 2017$?

Let $f_n(x)$ and $f(x)$ be continuous functions on $[0, 1]$ such that $\lim_{n\to\infty} f_n(x) = f(x)$ for all $x \in [0, 1]$. Answer each of the following questions. If your answer is “yes”, then ...
0
votes
1answer
37 views

$\lim_{n \to \infty}\left(\int_{a}^{b}\phi^{2n}\mathrm{d} \alpha \right)^{\frac{1}{2}} = \sup_{a \leq x \leq b} \phi^{2}(x)$

Problem. Let $\phi: [a,b] \to \mathbb{R}$ continous and $\alpha:[a,b] \to \mathbb{R}$ strictly increasing. Show that $$\lim_{n \to \infty}\left(\int_{a}^{b}\phi^{2n}\mathrm{d} \alpha \right)^{\frac{...
1
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0answers
38 views

Orthogonality of sine function: Discrepancy Between Lecture Notes and Textbook [on hold]

My lecture notes state that the orthogonality of the sine function is shown by the following integration: $$\int^1_0 g_0 \sin(m \pi x) \ dx = \sum_{n = 1}^\infty \int_0^1 A_n \sin(n \pi x) \sin(m \pi ...
5
votes
1answer
75 views

Integral of $\log(1-x^t)$ with respect to $t$

I would need some help to work with the following integral: $$f(x) = \int_2^\infty \log (1-x^t) dt ,\ \ \ \ \ \ \ \ \ |x|<1$$ I would like to get a closed form or something similar (which seems ...
-1
votes
0answers
26 views

Mean Value Theorem and Derivative Application Proof. Help please! [duplicate]

Suppose $f$ is a twice-differentiable function with $f(0) = 0$, $f\left(\frac12\right) = \frac12$ and $f'(0) = 0$. Prove that $f''(x) \ge 4$ for some $x \in \left[0,\frac12\right]$. So the hint that ...
1
vote
1answer
19 views

Volumes of revolution around the y-axis

How do I find the volume of the following function by using the disc method? I know it's easier with the shell method, but I'm required to do it with disc method and I'm missing something but can't ...
0
votes
0answers
20 views

Why does a new restriction suddenly appear when finding this potential function using antiderivatives?

I have 3 questions regarding the solution to the following problem: Let $$\vec{F} = M \hat i + N \, \hat j=\big(\sqrt{x^2+y^2}\big)^n (x \, \hat i + y \, \hat j)$$Whenever possible, find a ...
1
vote
1answer
39 views

How to calculate this Riemann-Stieltjes integral?

How to calculate this Riemann-Stieltjes integral? \begin{equation} \int_{1}^{3}e^{x}d\left\lfloor x\right\rfloor \end{equation} If $x\in\left[0,\,3\right]$, then $\left\lfloor x\right\rfloor =I\...
2
votes
1answer
43 views

Integration in normed vector spaces

Given an interval $I\subset \mathbb{R}$ and a normed vector space $X$, I want to know if I am able to define the Lebesgue space $L^p(I;X)$ of all $p$-integrable functions $f:I\to X$. I know that this ...
0
votes
0answers
17 views

Dirac's delta function with vector argument and relation to Dirac's delta of the vector's magnitude

Is there some way to approximately express Dirac's delta function with argument from $\mathbb{R}^2$ as delta function of magnitude of the vector, in other words, is there some way (under certain ...
4
votes
3answers
133 views

Evaluate: $\int_{0}^{\infty}\left(x^2-3x+1\right)e^{-x}\ln^3(x) dx$

$$I=\large \int_{0}^{\infty}\left(x^2-3x+1\right)e^{-x}\ln^3(x)\mathrm dx$$ $$e^{-x}=\sum_{n=0}^{\infty}\frac{(-x)^n}{n!}$$ $$I=\large \sum_{n=0}^{\infty}\frac{(-1)^n}{n!}\int_{0}^{\infty}\left(x^2-...
4
votes
0answers
58 views
+50

Choosing a boundary for integration

I have the following differential equation $$ \frac{d}{dx}\left(\mu e^{cx}f(x)\right) = -\mu\left(\frac{a xe^{-cx}}{a x+x-1}\right) $$ that I am trying to integrate to find $f(x)$ with the boundary ...
-2
votes
2answers
36 views

Proving an entire function with constant imaginary part on closed unit disc is constant on the whole $\mathbb C$ without identity principle

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Exer 8.24 (Exer 8.24) If $f: \mathbb C \to \mathbb C$ is entire and $\Im(f)$ is constant on ...
2
votes
1answer
61 views

Evaluate $\int_0^{\frac{\pi}{2}} \frac{1}{\sqrt{a^2 \cos^2 \theta+b^2 \sin^2 \theta}} d \theta$

$\int_0^{\frac{\pi}{2}} \frac{1}{\sqrt{a^2 \cos^2 \theta+b^2 \sin^2 \theta}} d \theta$ $ = \int_0^{\frac{\pi}{2}} \frac{1}{a}\sec \theta \frac{1}{ \sqrt{1+(b/a)^2 \tan^2 \theta}} d \theta$ But i ...
0
votes
1answer
37 views

Prove path independence of $\int_{\gamma} f$ w/ weaker conditions on $f$

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Ch5.2 2 Questions about Cor 5.8 and Cor 5.9 (*) Question 1. Can we prove Cor 5.9 using Cor 5....
3
votes
1answer
46 views

Order and principal part of $f(x)=\, e^{x^2} \int_{x}^{+\infty}e^{-t^2}\,dt$, infinitesimal as $x\to+\infty$

Determine if $f(x)$ is an infinite or an infinitesimal as $x\to+\infty$, its order and the principal part $$f(x)=\, e^{x^2} \int_{x}^{+\infty}e^{-t^2}\,dt$$ I can write, and then, using L'Hôpital's ...