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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11 views

Matrix Multiplication, Trace and Integration

Let $\omega(x)$ be a $p\times 1$ vector-valued function defined on a random variable $X$ with CDF $F$. Now define $$V:=\int \omega(x)[\omega(x)]^T dF(x).$$ Then define $\gamma$ as follows. $$ \gamma ...
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0answers
5 views

Primitive-ability of a function

Prove that the function $f:{R}\to{R}$ is primitive-able(does this term exist in English?) and find one of its primitives. $$f(x)=\left\{\begin{array}{cc} 1-x & x<1 \\ x^2-2x+1 & x \geq 1 ...
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1answer
32 views

Problem about integration

Let $\mathcal R$ be a $\sigma$-algebra in a nonempty set $X$, let $\mu$ be a positive measure on $\mathcal R$, let $f:X\to \mathbb C$ be measurable relative to $\mathcal R$,and $f\in L^1(\mu)$. Let ...
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2answers
27 views

How to derive ∫f'(x)[f(x)]dx

I need to show how ∫f'(x)[f(x)]dx is derived using integration techniques. Any specific examples that I can use to derive this formula? Thanks! Q: "Using clear explanations in the form of words and ...
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1answer
33 views

Closed form for an integral

I am trying to find a closed form for this integral: $\int\limits_{a}^{\infty} \exp(-\frac{b}{x})\exp(-cx)dx$ where a,b,c, are positive constants. Does anyone have any suggestions or can advise? ...
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0answers
11 views

Convergence of Laplace transform and its inverse

There is a sequence of functions $F^{\epsilon}(\lambda)$ which converges to 0 as $\epsilon \rightarrow 0$. Assume that each $F^{\epsilon}(\lambda)$ has a inverse Laplace transform f(s) such that ...
4
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0answers
44 views

Fredholm integral?

If one exists, find a continuous, bounded function $f: \mathbb{R} \to \mathbb{R}$ which is not identically zero and which satisfies$$0 = \int_0^\infty K(t, s)f(s)\,ds$$for all $t \in \mathbb{R}$, ...
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2answers
48 views

Show that $\int_0^\infty e^{-x}{\sqrt x}dx=\frac{\sqrt\pi}{2}$ by using $\int_0^\infty e^{-x^2}dx=\frac{\sqrt\pi}{2}$

I'm trying to do integration by parts to be able to use $\int_0^\infty e^{-x^2}dx=\frac{\sqrt\pi}{2}$, but is not working.
4
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2answers
52 views

What happens when I convert a Taylor series into an integral?

Suppose we have the Taylor series of an analytic function: $$f(x) = \sum_{k=0}^\infty \frac{1}{k!} a_k x^k$$ Then I decide to (kind of) turn it into an integral: $$g(x) = \int_0^\infty ...
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2answers
50 views

Prove convergence of $\int_1^\infty \frac 1 {x(\sqrt x + 1)} dx$

Prove the convergence of $\int_1^\infty \frac 1 {x(\sqrt x + 1)} dx$ This was a question on an exam. I needed to prove that the above integral converges using the comparison test. I thought about ...
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2answers
63 views

Does anyone know of a closed form solution to the following integral?

Does anyone know of a closed form solution to the following integral? $$ \DeclareMathOperator\erf{erf} \newcommand{d}{\;\mathrm{d}} \int^{+\infty}_{-\infty} \erf^{\;m}\!(x) \frac{\d^n ...
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1answer
26 views

Question about the limits of definite integrals

Let me take an example that I've come across while studying Fourier series, We all know that $$\int_{-a}^{a} \sin \left( \frac{n\pi x}{a} \right) dx = 2 \int_{0}^{a} \sin \left(\frac{n \pi x}{a} ...
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2answers
18 views

Space of all improper Riemann-integrable functions not closed under products and other operations

If $R[a,b]$ denotes the space of all Riemann-integrable functions in the closed interval $[a,b]$, then this space is closed under taking linear combinations, product of functions, powers of functions ...
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2answers
38 views

Limits and definition integrals involving logarithms

Let $a \in (0,1)$ and define $$I_n(a)=\int_a^1 (\ln x)^n \, \mathrm{d}x$$ Show that limit as $a\to 0$ we have, $$\lim_{a\to 0}I_n(a)=(-1)^n \cdot n!$$
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2answers
63 views

Role of i in Fourier transform

I've seen several derivations of the Fourier transform, but most don't cover the conversion to the form $$ S(f) = \int_{\infty}^{-\infty} s(t)e^{-i2\pi ft} \;\mathrm{d}t $$ What is the role of ...
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2answers
28 views

Numerical integration of divergent function

I am having trouble with the numerical integration of a divergent function. For example, \begin{equation} n= \int f(x)\,dx = \displaystyle\int \dfrac{\Theta(x-\varepsilon)\,dx}{\sqrt{x-\varepsilon}} ...
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1answer
41 views

How find $I= \int_{x=0}^{ \frac{1}{2} } \int_{y=x}^{1-x} ( \frac{x-y}{x+y})^{2}\, dy\,dx$

In $$I= \int_{x=0}^{ \frac{1}{2} } \int_{y=x}^{1-x} \left( \frac{x-y}{x+y}\right)^{2} \,dy\,dx$$ follow the change of variables on $x= \frac{1}{2} (r-s),y= \frac{1}{2} (r+s)$ and find$I$ My try ...
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0answers
10 views

Coefficients and synthesis of Associated Legendre Polynomials

First of all, all the Associated Legendre Polynomials (ALP) I'm mentioning below are NORMALISED according to the convention of Spherical Harmonics, and the ALPs can be accessed in Mathematica using ...
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3answers
297 views

How would I integrate the following?

I asked my Maths teacher recently how would you integrate the following, $$\int {x^x}^2 \, \mathrm{d}x$$ and she wasn't quite sure, I read you need to use as $x \to \infty$ but this was only briefly ...
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0answers
6 views

Numerical integration of divergent functions

I am having trouble with the numerical integration of a divergent function. For example, \begin{equation} n= \int f(x)\,dx = \displaystyle\int \dfrac{\Theta(x-\varepsilon)\,dx}{\sqrt{x-\varepsilon}} ...
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0answers
13 views

Is there a Gronwall-type lower bound inequality?

There are various versions of Gronwall's lemma. One of them is something like the following: If $f(t) \leq h(t) + \int_0^t g(s)f(s)ds$, plus some continuity conditions, then $f(t)\leq$ something ...
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1answer
30 views

Integrability of a function

Show that the function is integrable on $[0,2]$ $$f(x)=\left\{\begin{array}{cc} 1-x & x<1 \\ x^2-2x+1 & x \geq 1 \end{array}\right.$$ What conditions need to be checked in order for it to ...
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41 views

Is it possible to integrate this Riemann zeta function ratio so that I can produce this graph?

I am partly repeating myself here. But the form of this expression is nicer than the one I suggested here. I would like to integrate this: $$1-\frac{\zeta \left(\frac{1}{2}+i t\right)}{\zeta ...
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1answer
33 views

Integration of a generic radial function in polar coordinates

I need to perform the following integral $\int{P(k) e^{i \vec{k}\cdot \vec{\Delta r}} \frac{d^2k}{(2 \pi) ^2}}$ using polar coordinates. I think the result should depend on some Bessel function, but ...
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0answers
30 views

Convergence of a sequence of integration

I am considering one problem and I am stuck in this step. The problem is that What conditions on function $f(u,\epsilon)$ are required to satisfy $$ \int_0^\epsilon f(u,\epsilon)\,du \rightarrow 0 ...
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1answer
50 views

Evaluation of the integral $\int_0^1 e^{2t^2 -at} dt$

I would like to integrate a function in the range $[0,1]$. I tried a lot of ways including Mathlab. All solutions come in terms of some error function. I would like the answer in terms of $a$. ...
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3answers
40 views

Evaluating the closed integral of an elliptical path

I've been working on a problem that states: Evaluate $\int F*dr $ where $F(x,y,z) = x\,i+xy\,j+x^2yz\,k $ and C is the elliptical path given by $$ x^2+4y^2-8y+3=0 $$ in the xy-plane, traversed ...
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77 views

I have great doubts solve this exercise by integral by parts $\int_{0}^1 \int_0^1 x\cdot e^{xy}\, dy\, dx$ [on hold]

I have great doubts solve this exercise by integral by parts $\int_{0}^1 \int_0^1 x\cdot e^{xy}\, dy\, dx$
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1answer
16 views

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the y-axis

I am having a little trouble figuring out how to integrate this problem. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the y-axis. ...
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1answer
22 views

Finding the surface area of the solid formed by a revolution of the function $f(y)=x$ when rotated about the line $y=0$.

I know of the following formulas for calculating surface areas: $\displaystyle A_S = 2\pi\int_{a}^{b}f(x)\sqrt{1+f'(x)^2}{\ dx}$ for the surface area ($A_S$) of the solid formed by revolving $f(x) = ...
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0answers
30 views

Lebesgue-Stieltjes: Computation

Problem Given the real line $\mathbb{R}$. Consider a Borel family: $$\mu(\mathbb{R})<\infty:\quad\mu(\lambda):=\mu(-\infty,\lambda]$$ How can I compute: ...
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2answers
95 views

How to calculate $\int \frac{\sin x}{\tan x+\cos x} \, dx$

How to calculate $$\int \frac{\sin x}{\tan x+\cos x} \, dx\text{ ?}$$ I got to $$\int \frac{-u}{u^2-u-1} \, du$$ while $u=\sin x$ but can I continue from here?
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1answer
47 views

Assumptions on functions so that integral is zero

Let $f:\mathbb{R}\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ be two arbitrary functions. Assume $g\in L^2(\mathbb{R})$. I'm looking to find out the minimal set of assumptions on $f$ and $g$ such ...
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1answer
33 views

find the minimum value of this integral when $1>t>0$, $f(t)=\int\limits_0^1 x |e^{-x^2} - t|\ \mathrm{d}x = ?$

Is there someone who can show me How do i find the minimum value of this integral when $1>t>0$, \begin{align*}f(t)=\int\limits_0^1 x |e^{-x^2} - t|\ \mathrm{d}x &= \end{align*} Note : ...
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1answer
41 views

convolution and integral limits

Let $\xi$ be an increasing function , and $f$ be a continuous function on the interval $[0,1]$. Take $\phi$ a smooth function such that $\int_0^1 \phi(s)\, ds= 1 $ and consider an approximation of ...
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1answer
20 views

Asymptotic behaviour of Hilbert transform

Let $f$ be a bounded function on $\mathbb{R}$ with compact support include in $[-K,K]$. Show that $$ H(f)(x)=\frac{a}{\pi x}+O(\frac{1}{x^2})$$ where $a=\int f(t)dt$ and $H$ denote the Hilbert ...
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0answers
25 views

Change of variable in double and triple integrals?

I learn double and triples integral as same as change of variable and then surface integral in my class so there is some conflict between how to do double integrals Here is how the text book say ...
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3answers
63 views

Does $\int_a^\infty f$ exist iff $\int_a^\infty |f|$ exists?

My question is, does $\int_a^\infty f(x)dx$ exist if and only if $\int_a^\infty |f(x)|dx$ converges? Since $$\left|\int_a^\infty f(x)dx\right|\leq \int_a^\infty |f(x)|dx,$$ it's obvious that if ...
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1answer
57 views

Does $\int_0^\infty \frac{1}{1+(x\sin x)^2}\ dx$ converge?

Does the integral $$\int_0^\infty \frac{1}{1+(x\sin x)^2} \ \, \mathrm{d}x$$ converge? I know that I need to look at: $$\sum_{n=0}^\infty \int_{n\pi}^{(n+1)\pi} \frac{1}{1+(x\sin x)^2}\ \, ...
3
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1answer
35 views

Proving that the Gamma function $\Gamma(y)$ converges for $y>0$.

How can I justify that $$\Gamma(y)=\int_0^\infty t^{y-1}e^{-t} \, \mathrm{d}t$$ exists for all $y>0$? I'm struggling to compare it to a known convergent integral.
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1answer
27 views

Every step function is a linear combination of elementary step functions.

If $J$ is any subinterval of $[a, b]$ and if $\phi_J (x) := 1$ for $x \in J$ and $\phi_J (x) := 0$ elsewhere on $[a, b]$, we say that $\phi_J$ is an elementary step function on $[a, b]$. Then to ...
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2answers
36 views

Proving $\int_0^1 \frac{f(t)}{t^{\alpha + 1}} \ dt$ diverges

Consider $f(t)$, continuous on $[0,1]$, and $\alpha > 1$, and: $$\int_0^1 \frac{f(t)}{t^{\alpha + 1}} \ dt$$ How can we tell this integral diverges? Basically since $f$ is continuous it reaches ...
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1answer
37 views

Is my proof of closedness of multiplication operator corect?

I am considering an operator $A: L^2(\mathbb R , d \mu) \supset D(A)\to L^2 (\mathbb R, d\mu)$ defined by $(Af)(x)=a(x)f(x)$ for known measurable function $a$. Domain is of course all those functions ...
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6answers
76 views

How to evaluate this integral $\int\limits_1^4\!\left( \frac{1}{\sqrt{x}}+\frac{1}{x}\right) \mathrm{d}x $?

$$\int_1^4\!\left( \frac{1}{\sqrt{x}}+\frac{1}{x}\right) \mathrm{d}x $$ The answer is $2+\ln(4)$, however I don't understand why. What I did was the following: $$\ln(x^{0.5})+\ln(x) = ...
2
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1answer
38 views

Help with Definite integral question

Anyone please help with this question: (a) Show that: \begin{align} \int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-x) dx \end{align} (b) Hence show that: \begin{align} \int_{0}^{\frac{\pi}{4}} ...
4
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4answers
410 views

Why can we treat infinitesimals as real numbers in integration by substitution?

During integration by substitution we normally treat infinitesimals as real numbers, though I have been made aware that they are not real numbers but merely symbolic, and yet we still can, apparently, ...
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1answer
46 views

Finding the general integrals of functions like $\frac1{x^n+1}$, $\cos^nx$. [on hold]

This question is just a soft question, about can we compute a general formula for everything? Or it has some restrictions? Like $\int x^ndx=\frac{x^{n+1}}{n+1}+C$. I am not able to deduce a formula ...
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1answer
46 views

Help understanding proof on Jensen's Inequality

I need help understanding the proof for Jensen's inequality in "Real and Complex Analysis" by Rudin. 3.3 Theorem (Jensen's Inequality) Let $\mu$ be a positive measure on a $\sigma$-algebra ...
2
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3answers
131 views

How do i evaluate this integral $ \int_{\pi /4}^{\pi /3}\frac{\sqrt{\tan x}}{\sin x}dx $?

Is there some one show me how do i evaluate this integral :$$ \int_{\pi /4}^{\pi /3}\frac{\sqrt{\tan x}}{\sin x}dx $$ Note :By mathematica,the result is : $\frac{Gamma\left(\frac1 ...
2
votes
3answers
46 views

Continuity of function consisting of an infinite series.

Let $f(x) , 0\leq x\leq 1$ be defined by, $$f(x)=\sum_{n=1}^{\infty}\frac{1}{(x+n)^2}$$. Show that $f$ is continuous on $[0,1]$ and that, $$\int_0^1f(x)dx=1$$. I have never dealt ...