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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1answer
25 views

Finding fomulas for hyperbolic functions

I'm trying to find formulas for hyperbolic functions, starting with this image Knowing that the area between the origin, vertex and a point on hyperbola (enclosed by x-axis and hyperbola itself) is ...
1
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4answers
43 views

How does this algebraic trick regarding partial fraction works?

Suppose I have to evaluate the integral $$\int \frac{x}{(x-1)(2x+1)(x+3)} \, dx $$ I write it as $$\frac{a_1}{x-1} +\frac{a_2}{2x+1} +\frac{a_3}{x+3}$$ where $a_1$, $a_2$, $a_3$ are constants. once I ...
1
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0answers
19 views

Is $X(T) = A \sin(\omega_0 t + \Phi)$ mean ergodic?

This is an example of a tutorial but I think has not been solved properly. Please help me! $X(T) = A \sin(\omega_0 t + \Phi)$ $A$ and $\phi$ are independent $A$ is uniformly distributed over ...
1
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2answers
30 views

how they prove this Fourier Transform of unit impulse function

In my text book , to prove $\ \frac{1}{2\pi}\int_{-\infty}^{\infty}e^{j\omega t}d\omega $ behaves like unit impulse function they evaluate the integral : $\ ...
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0answers
16 views

Integral: Product of two Poisson integrals

$\textbf{Problem}$: Compute the integral $$ I(x) = \int_{-\infty}^0 \phi_x^2+\phi_z^2 \ dz $$ where $$\phi_x = P.V.\frac{1}{\pi} \int_{-\infty}^{\infty} \frac{\partial (g(y)^2)}{\partial y} ...
0
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1answer
45 views

Find the integral of $x(x^2+4)^5$? [on hold]

Please do show me step by step way of integrating $x(x^2+4)^5$. Please do provide me with the formulas, if any. In my text-book, the answer is given as: $(\frac{1}{12})(x^2+4)^6 + k$ But I don't know ...
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4answers
45 views

Explanation of this integral's solution

While doodling around with circles and associated geometry, I've stumbled across this integral (I'm led to believe it is correct but I have not found or created any proof): ...
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0answers
6 views

Numerical integration of functions over computable Cauchy sequences

I'm interested in exact real arithmetic (and by extension constructive analysis). A nice representation of real numbers is via Cauchy Sequences. The basic idea being that you have a function which, ...
5
votes
1answer
54 views

Evaluate integral with gaussian curvature

I thought evaluating it in the following way: $$\begin{align} \int_0^{2\pi}\int_0^{\pi}K(x,y)\sqrt{\det(g_{ij})} \, dy\,dx &= \int_0^{2\pi}\int_0^\pi \sqrt{\det L_{ij}}\cdot \sqrt{{\frac{\det ...
7
votes
1answer
76 views

What exactly IS a line integral?

As what happens in many math courses, a topic is learned without truly learning what one is doing. For me, this is line integrals. I can do them well, I just never truly learned what exactly I was ...
5
votes
3answers
67 views

Closed-form of $\int_0^1 x^n \operatorname{li}(x^m)\,dx$

I've conjectured, that for $n\geq0$ and $m\geq1$ integers $$ \int_0^1 x^n \operatorname{li}(x^m)\,dx \stackrel{?}{=} -\frac{1}{n+1}\ln\left(\frac{m+n+1}{m}\right), $$ where $\operatorname{li}$ is the ...
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0answers
16 views

analytical solution of non-linear least square problem

I am implementing a trust region optimization algorithm and I would like to compare it against already done similar work, where authors measures performance on this problem. $$ \min_{u,\gamma}\Bigg\{ ...
1
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2answers
48 views

For what $(a,b) \in R^+$ does $\int^\infty_b (\sqrt{\sqrt{x+a}-\sqrt{x} \vphantom{\sqrt{x}-\sqrt{x-b}}}-\sqrt{\sqrt{x}-\sqrt{x-b}})dx$ converge?

For what pairs $(a,b) \in R^+$ does this integral converge? $$ \int\limits^{\infty}_{b} \left (\sqrt{\sqrt{x+a}-\sqrt{x} \vphantom{\sqrt{x}-\sqrt{x-b}}}-\sqrt{\sqrt{x}-\sqrt{x-b}} \right)dx $$
0
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0answers
18 views

Arc Length for Superposition of Sinusoidal Curves

I am wanting to compute the arc length, $s$, of a superposition of two sinusoidal functions--say $$y(x) = A\cos\left(k_1 x\right)+B\cos\left(k_2 x\right).$$ There is a special relationship between ...
2
votes
1answer
31 views

Fourier transform of $1-\frac{|\tau|}{2T}$

So far I have tried the following: $$\begin{align} \mathscr{F}(f)&=\mathscr{F}\{1-\frac{|\tau|}{2T}\}\\ &=\int_{-\infty}^{+\infty}(1-\frac{|\tau|}{2T})e^{-i\omega\tau}d\tau\\ ...
2
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5answers
36 views

Finding a general solution to a differential equation, using the integration factor method

Use the method of integrating factor to solve the linear ODE $$ y' + 2xy = e^{−x^2}.$$ And verify your answer I can solve the ODE as a linear equation (mulitply both sides, subsititute, reverse ...
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2answers
25 views

Problem with understanding the application of the Intermediate Value Theorem in the proof of the Mean Value Theorem for Integrals

I am struggling to understand the last parts of this proof because I know that the IVT states that on the interval $[a,b]$ of $f$, where it is continuous, there exists a value $L$ between $f(a)$ and ...
1
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0answers
42 views

Is there a change-of-variables solution for integrals from negative infinity to a constant?

I found a fantastic and generalizable substitution technique for computing definite integrals that go to infinity from either negative infinity or a constant, regardless of the function (sorry for the ...
0
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1answer
30 views

$f(x)=2-|x-3|, 1\le x\le 5$ and for other values, $f(x)$ is obtained using the relation $f(5x)=kf(x)$ for $x\in R$. then…

Question: The maximum value of f(x) in $[5^4,5^5]$ for $k=2$ is? Also, if $$\lim_{x\to \infty}\int_1^xf(x)dx$$ is a finite number, find the exhaustive set of $k$. Attempt : For first part, ...
0
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1answer
31 views

Dividing a circle's area into fourths via parallel lines

My attempt at this solution involves first finding the equation of a half circle $x=\sqrt{(r^2-y^2)}$, $\int_0^mf(y)dy=\frac {\pi}{8r^2}$ Is there an easier solution? My attempt requires ...
1
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1answer
24 views

what step should i choose to get needed precision in trapezoidal method

What step should i choose to get needed precision in trapezoidal method? ...
1
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0answers
22 views

Conditions on $f$ to have $ \int_{x=0}^1\int_{y=0}^1\int_{z=0}^1 \frac{f(x)}{(x-y)^2 (y-z)} dz dy dx $ finite?

Suppose that $f$ is a $\mathcal{C}^\infty$ function. $$ \int_{x=0}^1\int_{y=0}^1\int_{z=0}^1 \frac{f(x)}{(x-y)^2 (y-z)} dz dy dx $$ Which are the conditions on $f$ that makes this integral finite ? ...
9
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1answer
68 views

Closed-forms of the integrals $\int_0^1 K(\sqrt{k})^2 \, dk$, $\int_0^1 E(\sqrt{k})^2 \, dk$ and $\int_0^1 K(\sqrt{k}) E(\sqrt{k}) \, dk$

Let denote $K$ and $E$ the complete elliptic integral of the first and second kind. The integrand $K(\sqrt{k})$ and $E(\sqrt{k})$ has a closed-form antiderivative in term of $K(\sqrt{k})$ and ...
0
votes
2answers
52 views

Contour integration of exponential function [on hold]

How to solve this integral with residues method? $$\int_0^\infty \frac{e^{ixp}}{x^2+1+i}dx$$
2
votes
1answer
42 views

Finding a Solution to a linear Voltera equation of the second type

I want to solve the following integral equation: $$ u(t) = \int_t^T a(s) ds + \int_t^T b(s)u(s) ds , $$ with $a, b, u$ being functions from $[t,T] \rightarrow \mathbf{R} $. I transformed the ...
6
votes
2answers
523 views

A problem in integration.

As you know from basic trigonometry that $\sin(2x) = 2\sin(x)\cos(x)$. If you integrate both sides with respect to x, one finds $$\int \sin(2x) \ dx = -\frac{1}{2}\cos(2x)+c$$ on the left hand side ...
1
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0answers
34 views

Proof of additivity of domain for definite integrals

I would like to prove the following theorem: Theorem If $c \in (a,b)$ and $f$ is integrable on $[a,c]$ and $[c,b]$, then $f$ is integrable on $[a,b]$ and $$\int_{a}^{b}f = \int_{a}^{c}f + ...
-3
votes
2answers
61 views

Analytic integration of this function [on hold]

Integrate \begin{equation} \int{\frac{1}{(1-\frac{a}{r}-b r^2)}} \, \mathrm{d}r \end{equation} where $a$ and $b$ are constants.
1
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1answer
36 views

Integral related to Poisson kernel

$\textbf{Problem}$: Find the value of the integral $$I=\int_{-\infty}^0 P.V.\frac{1}{\pi}\int_{-\infty}^{\infty} \frac{\partial f}{\partial y} \frac{(x-y)}{(x-y)^2+z^2} \ dy \ dz,$$ with $f$ a ...
3
votes
4answers
89 views

Expressing the integral in terms of the original variable

In evaluating the integral: $$ \int{dx\over(a^2-x^2)^{3/2}} $$ or $$ \int{dx\over(a^2-x^2)^{1/2}\ (a^2-x^2)}$$ Let $ x=a\sin\theta $ and $ dx=a\cos\theta\ d\theta $. Then $$ \int{{a\cos\theta\ ...
5
votes
5answers
1k views

Math Subject GRE 1268 Question 55

If $a$ and $b$ are positive numbers, what is the value of $\displaystyle \int_0^\infty \frac{e^{ax}-e^{bx}}{(1+e^{ax})(1+e^{bx})}dx$. A: $0$ B: $1$ C: $a-b$ D: $(a-b)\log 2$ E: ...
2
votes
1answer
53 views

Inequality on integrals of continuous functions: $\int_0^1 f^2(x)\,dx \geq \left(\int_{0}^{1} f(x) \,dx\right)^2$

Let $f\colon [0, 1] \to \mathbb{R}$ be a continuous function. How to prove $$\int_0^1 f^2(x)\,dx \geq \left(\int_{0}^{1} f(x) \,dx\right)^2$$ (I'm not getting anything.. any hint is appreciated)
2
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0answers
39 views

Problem with a step involving a type of Riemann integration

I am reading this text, , and I find it unclear how the ratio of the considered rectangle's area to its length tends to become the derivative of a function $S$ as the lenght of the considered ...
2
votes
3answers
47 views

compute temporal average of $\sin(\omega_0t+\Phi)\sin(\omega_0t+\omega_0\tau+\Phi)$

assuming that $\Phi$ is uniformly distributed over $(0,2\pi)$ compute: $$E[\sin(\omega_0t+\Phi)\sin(\omega_0t+\omega_0\tau+\Phi)]$$ I have solved the problem as continues: $$\begin{align} ...
2
votes
0answers
59 views

An integral that I cannot simplify.

Good day, esteemed students of mathematics! I have been trying to prove that the convolution of $2q$ Gaussian probability distributions is another $q$ Gaussian probability distribution with the same ...
1
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0answers
77 views

Integrating functions with $x^3$

After learning the integration of various functions with $x^2$ involved, I was given the following integration, as a challenge: $$\sqrt{1+x^3}$$ I tried various methods - too long to even try and ...
2
votes
2answers
65 views

Is $\int_{x=0}^1\int_{y=0}^1\int_{z=0}^1 \frac{1}{(x-y)^2 (y-z)} dx dy dz$ finite?

My question is in the title : How could I prove that $$ \int_{x=0}^1\int_{y=0}^1\int_{z=0}^1 \frac{1}{(x-y)^2 (y-z)} \ \text{d}z \ \text{d}y \ \text{d}x $$ is finite (if it is) ? Thank you by ...
0
votes
2answers
66 views

Compute the integral over the volume of a torus,

In $\mathbb R^3$, let $C$ be the circle in the $xy$-plane with radius $2$ and the origin as the center, i.e., $$C= \Big\{ \big(x,y,z\big) \in \mathbb R^3 \mid x^2+y^2=4, \ z=0\Big\}.$$ Let $\Omega$ ...
1
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2answers
33 views

Evaluating the bounds for a triple integral

I've working on the problem: Evaluate $\iiint_Q$ $1/(x^2 + y^2 + z^2)$, where Q is the solid region ABOVE the xy-plane (and we must do this in spherical coordinates). What I've done thus far is ...
0
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0answers
37 views

Confused on what the question is asking… Integration and Riemann sums

So I know how to integrate, and I understand Riemann sums via Right hand, left, etc but this question I am trying to answer has me baffled. I think its because its trying to be generic in the f(mx) ...
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votes
1answer
49 views

Is there an equation for the integral? [on hold]

For example, the equation of a derivative would just be f(x+h)-f(x)/h What would the equation of an integral be? Or does it just go backwards.
1
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0answers
25 views

Will this method always find the maximum of a positive-definite function?

Definitions: A real-valued, continuously differentiable function $f$ is positive definite on a neighborhood of the origin, $D$, if $f(0) = 0$ and $f(x) > 0$ for every non-zero $x \in D$. ...
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votes
0answers
22 views

How do you calculate an area enclosed by four tangents by using the integration method?

For example, make it $y=3x-6$, $y=3x-15.48$, $y=-0.25x+1.25$, and $y=-0.25x-1.06$. It's been taken by finding the tangent line of a curve $y=(x-2)(x-3)(x-5)$.
0
votes
1answer
34 views

To what fractional Sobolev spaces does the step function belong? (Sobolev-Slobodeckij norm of step function)

I'm new to fractional Sobolev spaces and I'm curious about the regularity of some simple functions like e.$\,$g. step functions in order to understand these spaces better. In more detail, for $\Omega ...
4
votes
0answers
78 views

Integral $\int z^2\Re(J_1(z))dz$=$\int y^{3/2} \Re \left[\frac{1}{\sqrt y} (1-e^{-y})\right]dy$

Hi I am trying to simplify and calculate the integral below. $$ I=\int x^2 \, \Re\left[{J_1(a x)}\right]dx=\frac{1}{a^3}\int z^2 \Re\left[\frac{z}{2}\sum_{k\geq 0} \frac{(-1)^k}{k!\Gamma(k+2)} ...
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votes
1answer
59 views

Integrations Question Answer [on hold]

I need your help to evaluate this integral: $$\int\frac {x^5}{1+x^6}~\mathrm{d}x $$ Any help will be appreciated!
12
votes
1answer
171 views

Could it possibly have a nice closed form? $\int _0^1\int _0^1\frac{x y}{(x+1) (y+1) \log (x y)}\ dx \ dy$

Using multiple integrals it's not hard to show that the present integral reduces to some integral over squared digamma functions, but then things become harder. How would you tackle the problem? ...
1
vote
1answer
39 views

integrating product of PDF and CDF

I am trying to show that the following integral: $$ \int_{-\infty}^a F(x)~f(x)~dx = \frac{F(a)}{2!} $$ Where $F$ is the cumulative distribution function of some continuous random variable X, and $f$ ...
5
votes
3answers
121 views

Problem 7 IMC 2015 - Integral and Limit

I'm trying to solve problem 7 from the IMC 2015, Blagoevgrad, Bulgaria (Day 2, July 30). Here is the problem Compute $$\large\lim_{A\to\infty}\frac{1}{A}\int_1^A A^\frac{1}{x}\,\mathrm dx$$ ...
-5
votes
2answers
39 views

solve this linear equation

Using linear differential equation, solve the following equation $( y \log (x)-2) y \textrm{d} x =x \textrm{d}y$. Source: "higher engineering mathematics by grewal"