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
9 views

Example of a non square-integrable martingale?

Are there (simple) examples of martingales which aren't square integrable?
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0answers
19 views

use L1-convergence to show integral convergence

Let $f\in L^1([0,1])$, $g_n$ a sequence of continuous functions that converges in $L^1$ to some $g\in L^1([0,1])$. Now my question is: Does $\int_0^1 f(t)e^{g_n(t)} dt$ converge to $\int_0^1 ...
1
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1answer
38 views

Calculate $I=\int_0^{1}\frac{1+x}{x^2+x+1}\operatorname{Log}\left({\frac{x}{1-x}}\right)\,d.x$ without using complex analysis

Calculate $$I=\int_0^{1}\frac{1+x}{x^2+x+1}\operatorname{Log}\left({\frac{x}{1-x}}\right)dx$$ without using complex analysis. How to calculate without using the residue theorem? The correct ...
2
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1answer
16 views

Which inequality are there with stochastic integration?

Which inequalities can I use with stochastic integration? For example, with the standard lebesgue integral we have $$\left|\int_\Omega f(x) dx\right| \le M |\Omega|$$ (where $M$ is the maximum of ...
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1answer
19 views

Changing the order of a double integration ? $\int_{-5}^{5}dx\int_{-7}^{\sqrt{25-x^2}}f(x,y)dy$

I've been doing an example of changing the order of a double integral and I'm not sure if I did it right. I would really appreciate if someone would check if my solution is right and correct any ...
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1answer
32 views

How to show that this is a martingale?

Let $H_s$ be a predictable and bounded process. How can I show that $$M_t = \int_0^t H_s \, dW_s$$ is a martingale? Clearly since $H_s \in L^2_\text{loc} (W)$ we have that $M_t$ is a local ...
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0answers
20 views

To what Sobolev space does this function belong to?

I am given this function: $$f(x) = e^{- \sqrt{|x|}}$$ and I want to find $k\in \mathbb{N}, \ p \ge 1$ such that $f \in W^{kp} (\mathbb{R})= \{ f \in L^p (\mathbb{R}) \ | \ \forall \alpha \le k: \ ...
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1answer
24 views

Integrating this norm

The norm \begin{equation} ||u||^{2} = \int_{\mathbb R} 1 \cdot u(x) \cdot \overline{u(x)} dx \end{equation} Claim which should be correct \begin{equation} ||u||^{2} = \int_{\mathbb R} 1 \cdot u(x) ...
2
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3answers
62 views

Integral of $\int_{\mathbb{R}}e^{-\frac{x^{2}}{2}}\left(\cos\left(\pi nx\right)\right)dx$

I was in need to urgently solve this integral. I already know the result in the closed form, does anybody know how to solve it? \begin{equation} ...
1
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2answers
48 views

Integrating triangle in a 2D plane

I am interested in integrating $(x^2y+y^2x)$ on the following loop: $(x=1,y=2)\rightarrow(x=2,y=1)\rightarrow(x=3,y=3)\rightarrow(x=1,y=2)$. I know this loop forms a triangle with all three sides ...
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1answer
26 views

When should we use absolute value by solving this integral?

I have for example: $\int_{y_0}^{y} \frac{1}{2\eta} d\eta$ with $t_0$ a real constant and the solution was: $\frac{1}{2}(ln|y|-ln|y_0|)$ But on other case I had: $\int_{y_0}^{y} \frac{1}{\eta} ...
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1answer
17 views

Complex integral over line, similarity with conservative field

Have $\int_C(4z^2-2iz)dz$ integral. Does it depend on choice of path? Tried to express $f(z)=(4z^2-2iz)$, then $f(x+yi)=(4x^2-4y^2+2y)+i(8xy-2x)$ Then $\frac{\delta P}{\delta y}=-8y+2$ And ...
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0answers
26 views

Does this function exist an inverse function?

Could I find the inverse function of the following integral equation? I am going to write it as $h(x)=...$ The integral equation is: $$R(i) = \int_{y\in S} {\frac{{w(y) g(y,i)_{}^\sigma ...
4
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2answers
69 views

Find the integral $\int \frac{1+x}{\sqrt{1-x^2}}\mathrm dx$

The integral can be represented as $$ \int \frac{1+x}{\sqrt{1-x^2}}\mathrm dx= \int \left(\frac{1+x}{1-x}\right)^{1/2}\mathrm dx $$ Substitution $$t=\frac{1+x}{1-x}\Rightarrow ...
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1answer
18 views

Finding the center of mass for a centroid without a convenient symmetry axis

Find the centroid of the lamina described in polar coordinates as $\left \{ \strut \left ( x,y \right )~|~0\leq r\leq 4 \cos\left ( \theta \right ),0\leq \theta \leq \frac{\pi}{3} \right \}$ Having ...
2
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1answer
18 views

Calculate the surface integral of bounded cylinder

Evaluate $$\int\int z^2\,dS,$$ where $S$ is the part of outer surface of cylinder $x^2+y^2=4$ between the planes $z=0$ and $z=3$. The answer given in book is $\pi$ but I am not getting this ...
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1answer
29 views

Riemann integrability given by limit of $\frac 1n \sum_{k=1}^n[f(k/n)]$

If $f:[a,b] \to \mathbb R$ is such that $$\lim_{n \to \infty} \frac 1n \sum_{k=1}^n[f(k/n)] = 1,$$ does that imply that $f$ is Riemann integrable on $[a,b]$ ? Thank you.
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0answers
16 views

I need some referenced materials about such type of integral.

I am struggling with the following type of integral. I can't find any referenced materials of it. Could you recommend some for me? Any books, papers or lecture notes are ok. Thank you. $$R(i) = ...
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2answers
26 views

Find the area bounded by function $y^2=16-2x$, the tangent to the curve at the point $(6,2)$ and the $y$-axis

First we find a tangent line of function $y^2=16-2x$ at $T(6,2)$: $y_t-y_0=f'(x_0)(x-x_0)$ where $x_0=6,y_0=2$ $y^2=16-2x\Rightarrow y=\sqrt{16-2x}$ or $y=-\sqrt{16-2x}$ Derivative of ...
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0answers
29 views

Why would an equation switch signs when something becomes independent of time? (Traffic Flow)

EDIT: I'm too tired for math and the answer to my question is explained in a comment below. Should this post be removed? Not sure if it adds much to the community given that it was all just me ...
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0answers
24 views

Baby Rudin theorem 6.16, explanation that a Riemann Stieltjes integral could be expressed as a infinite series.

The theorem says: Suppose $c_n \geq 0$ for $1,2,3 ...$. $\sum c_n$ converges, $\{s_n\}$ is a sequence of a distinct points in $(a,b)$, and $\alpha (x) = \sum^{\infty}_{n=1} c_n I(x-s_n)$. Let $f$ be ...
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0answers
36 views

I have a question about calculating integral

I need some help to solve this integral: $$\int_{4x}^{\infty} \frac{w^{\frac{m}{2}}e^{-a\sqrt{w+2\sqrt{xw}}}}{\sqrt{w^2-4xw}}\mathrm{d}w.$$ Thank you
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0answers
8 views

Gaussian weighted intergal of Product of Gaussians

I'm trying to find a solution to the following function, My understanding is that the resultant function should still be a Gaussian, however I would like to define it as a linear function the ...
1
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1answer
23 views

Finding distance of $h(t)=t$ from a closed subspace $Y$ of $\pi$-periodic functions in $L^2(-\pi,\pi)$

Let $Y=\{f\in L^2(-\pi,\pi):f(t-\pi)=f(t) \text{for almost all $t\in(0,\pi)$} \}$ Show that there exists $g\in Y$ such that $$\|h-g\|_2=\inf \{\|h-f\|_2:f\in Y\}$$ where $h(t)=t$. Compute ...
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0answers
8 views

Relationships for estimating the value of a double integral with sample points

I have a double integral with function (x+y) where I am supposed to estimate the value of the double integral by using a Reimann sum with m = 2 and n = 3. For sample points, I must use lower left ...
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0answers
34 views

Show $\frac{y-x}{(2-x-y)^3}$ is not integrable on $[0,1]\times[0,1]$, not invoking Fubini's theorem.

The double integral $$I = \int_{[0,1]\times[0,1]}\frac{y-x}{(2-x-y)^3} dxdy$$ does not have a finite value. The two iterated integrals have different values (Counterexample to Fubini?). Then Fubini's ...
2
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3answers
67 views

How do I finish this trig integral $\int_0^{\pi/4}\frac{\sin^2 \theta}{\cos \theta}d\theta$?

I got up to the part where it's $$\frac{9}{125}\int_0^{\large \frac{\pi}{4}}\frac{\sin^2\theta}{\cos\theta}\,\,d\theta$$ but I can't figure out how to finish it off. By the way the original problem ...
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1answer
24 views

Help to evaluate integral in cartesian and cylindrical

I want to solve $$\iint_{R} (x+z)dR$$ where R is the first octant of the cylinder $x^2+y^2=9$ and between $z=0$ and $z=4$ I think it could be done in either cylindrical or Cartesian. I am having ...
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0answers
21 views

Volume of Revolution (Semi-Circle, Line)

This is purely for academic curiosity and is not part of any homework assignment, quiz, or exam. Suppose $R$ is the region bounded by the curves $x^2 + y^2 = 36$ on $[-6,0]$, $y = 2x-6$ on $[0,6]$ ...
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0answers
31 views

An integration problem relating to central limit theorem [on hold]

Here is a question regarding integration. Assume that $X_{1}$...$X_{n}$ are $n$ independent and identically distributed (i.i.d) random variables from unknown distribution function $F(x)$ with mean $0$ ...
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1answer
57 views

integrate $\int dx \frac{2x+1}{(x^2-9)^\frac{5}{2}}$

$$\int dx \frac{2x+1}{(x^2-9)^\frac{5}{2}}$$ $x=\frac{3}{\sin\theta}$ $dx=\frac{3\sin\theta}{\cos^2\theta}d\theta$ $$\int d\theta ...
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0answers
102 views

Another way of doing integration

What's your option for calculating this integral? No full solution is necessary, it's optional as usual. Calculate $$\int_0^1 \frac{2 \zeta (3)\log ^3(1-x) \text{Li}_2(1-x) }{x}-\frac{2 \zeta (3) ...
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4answers
123 views

Is there any way to solve integral of $\sqrt{8-x^{2}}$ without using $\sin$ or $\cos$ formulas?

I was thinking about the following integral if I could solve it without using trigonometric formulas. If there is no other way to solve it, could you please explain me why do we replace $x$ with ...
1
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3answers
52 views

intergate $\int \frac{x}{(x^2-3x+17)^2}\ dx$

$$\int \frac{x}{(x^2-3x+17)^2}\ dx$$ $$\int \frac{x}{(x^2-3x+17)^2}\ dx=\int \frac{x}{((x-\frac{3}{2})^2+\frac{59}{4})^2}\ dx$$ $u=x-\frac{3}{2}$ $du=dx$ $$\int ...
2
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2answers
150 views

Integrate $x$ to the power $x$… to the power $x$… infinitely

This came across my mind, integrating $x$ to the power $x$ infinitely, I couldn't find anything on it. $$\Large \int x^{x^{x^{x\,\cdots}}} \, dx$$ How would you go about this?
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6answers
55 views

Simple integration just learning this application

I want to integrate a function as $f(x)=\sin^{-1}x$. What should be the proper method of doing it?
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3answers
55 views

integrate $\int \frac{dx}{(9+x^2)^2}$

$$\int \frac{dx}{(9+x^2)^2}$$ $x=3\tan\theta$ $dx=\frac{3}{\cos^2\theta}d\theta$ $$\int\frac{\frac{3}{\cos^2\theta}}{(9[1+\tan^2\theta])^2} \,d\theta = ...
0
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1answer
78 views

Solution of this definite integral?

I want to find the expression for the following integral $$\int_0^\infty\text{d}x\frac{e^{i k x}}{x}$$ I have tried deriving with respect to $k$, transforming into an integral over the whole real ...
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0answers
10 views

Cdf of truncated distribution

Let $X$ be a random variable with density $f_x$ and distribution function $F_x$. Define the interval $I = (a,b)$. Given that we know these and the inverse distribution function $F^{-1}_x$, how can we ...
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0answers
34 views

Integrating to find Action in Quantum Field Theory

I am struggling to show: $\int_{w=0}^{w'} \int_{r=2M}^{2(M-w)} \frac{-drdw}{1-\sqrt{\frac{2(M-w)}{r}-\frac{Q^2}{r^2}}}=2\pi[{2w'(M-\frac{w'}{2})-(M-w')\sqrt{(M-w')^2-Q^2)}+M\sqrt{M^2-Q^2}}]\\$ A ...
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2answers
29 views

trigo substitution and identites?

When I use trigo substitution to solve an integral I get an expression like that: $$\frac{1}{4}\tan\left(\arcsin\left(\frac{x-2}{2}\right)\right)+C$$ How can I simplify it?
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3answers
87 views

$\iiint_V \ x^{2n} + y^{2n} + z^{2n} \,dx\,dy\,dz$

$$\iiint_V \ x^{2n} + y^{2n} + z^{2n} \,dx\,dy\,dz$$ where V is the unit sphere. No information is given about n but I assume it is an integral. All I could think to do was to convert to spherical ...
3
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2answers
37 views

Intuition behind: Integral operator as generalization of matrix multiplication

So I am teaching myself more in-depth about integral operators and every once and awhile I see this little 'factoid', that integral operators are generalizations of matrix multiplications. In ...
2
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2answers
68 views

integrate $\int \frac{(16-9x^2)^{\frac{3}{2}}}{x^6}dx$

$$\int \frac{(16-9x^2)^{\frac{3}{2}}}{x^6}dx$$ $$\int \frac{(16-9x^2)^{\frac{3}{2}}}{x^6}dx=\int \frac{3\left(\frac{16}{9}-x^2\right)^{\frac{3}{2}}}{x^6}dx$$ $x=\frac{4}{3}\sin\theta$ ...
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1answer
52 views

Challenge in trignometry and integration [on hold]

Can anyone prove how the two equations are equal? Thanks $$=\frac1\pi \int_0^{2\pi} f(x) \left\{\frac12+\sum_{n=1}^N \cos [n(t-x)] \right\} \, dx$$ $$=\frac1{2\pi} \int_0^{2\pi} f(x) ...
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1answer
49 views

Evaluate $\int_{-\pi}^{\pi}\frac {\sin nx}{(1+2^x) \sin x} $

Evaluate $$\int_{-\pi}^{\pi}\frac {\sin nx}{(1+2^x)\sin x}dx \:\:\: n \in \mathbb{N}$$ $$\int_{-\pi}^{\pi}\frac {\sin nx}{(1+2^x)\sin x}dx = \int_{0}^{\pi}\frac {\sin nx}{(1+2^x)\sin x}dx + ...
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1answer
24 views

Approximating a Riemann integrable function using a continuous function

Let $f$ be Riemann integrable on $[a,b]$. Show that for every $ε > 0$, there is a continuous function $g$ on $[a,b]$ such that $$\int_a^b |f(x)−g(x)|\mathrm dx < ε. $$
3
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3answers
407 views

Integration with a constant “a”: $ \int_0^a \frac1{\sqrt {a^2-x^2}} dx $

Find the exact value of $$ \int_{0}^{a} \frac{1}{\sqrt {a^2-x^2}} \mathrm {dx} $$ Where, $a$ is a positive constant Hi, guys can give me tips to solve this ? Should we use like u substitution?
2
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0answers
35 views

How to prove that the following function has a unique mode?

I am trying to prove that the function $$f(\alpha)=n\ln \alpha-n\ln\Big(\sum_{i=1}^{n}t_i^\alpha+\int_{a}^{b}x^{\alpha+\beta-1}e^{-\lambda x^\beta}\,dx\Big)+(\alpha-1)\sum_{i=1}^{n}\ln t_i,$$ where ...
0
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0answers
14 views

Is there any trick for evaluate this integral?

Does the following function can be simplified or solved? $$R(i) = \int_{y\in S} {\frac{{w(y) g(y,i)_{}^\sigma }}{{\int_{x\in S} {h(x)g(x,y)_{}^\sigma f(x,y)_{}^\sigma dx} }}dy} $$ where S is a ...