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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5
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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: ...
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 ...
124
votes
2answers
8k views

Evaluate $ \int_{0}^{\frac{\pi}{2}}\frac1{(1+x^2)(1+\tan x)}\,\mathrm dx$

Evaluate the following integral$$\int_{0}^{\Large\frac{\pi}2}\frac1{(1+x^2)(1+\tan x)}\,\mathrm dx.$$ My Attempt: Let $$\tag1 I = \int_{0}^{\Large\frac{\pi}{2}}\frac{1}{(1+x^2)(1+\tan ...
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 ...
1
vote
0answers
18 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
vote
1answer
24 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 ...
21
votes
2answers
326 views
+200

A difficult logarithmic integral ${\Large\int}_0^1\log(x)\,\log(2+x)\,\log(1+x)\,\log\left(1+x^{-1}\right)dx$

A friend of mine shared this problem with me. As he was told, this integral can be evaluated in a closed form (the result may involve polylogarithms). Despite all our efforts, so far we have not ...
1
vote
4answers
42 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
vote
4answers
44 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): ...
0
votes
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} ...
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 ...
2
votes
2answers
106 views

Integration of $\int \frac{(1 + x)\sin x}{(x^2 +2 x)\cos^2 x-(1 + x)\sin2x}dx$

The integral is $$\int \dfrac{(1 + x)\sin x}{(x^2 + 2x)\cos^2 x-(1 + x)\sin2x}dx.$$I've tried the problem by first multiplying both the numerator and denominator by $\sec^2 x$ but couldn't do justice. ...
1
vote
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 : $\ ...
0
votes
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 ...
0
votes
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, ...
-1
votes
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
vote
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? ...
2
votes
3answers
383 views

Definite integrals using u substitution (verification needed)

Would someone mind verifying this? $ \int_{0}^{\ln(\pi + 1)}e^x \sin(e^x - 1) \space dx $ $ u = e^x - 1 \Rightarrow \frac{du}{dx} = e^x \Rightarrow du = e^x \space dx \Rightarrow dx = \frac{1}{e^x} ...
1
vote
1answer
33 views

How do I determine the new boundaries of $D ^* = T(D)$ when using change of variable?

I'm not quite sure how to complete this question $D$ is the region bounded by $x = 0, y=0,x+y=1$, and $x+y=4$. Using the change of variables $x = u -uv, y = uv$ and the Jacoian, evaluate the double ...
1
vote
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? $$ \int\limits^{\infty}_{b} \left (\sqrt{\sqrt{x+a}-\sqrt{x} \vphantom{\sqrt{x}-\sqrt{x-b}}}-\sqrt{\sqrt{x}-\sqrt{x-b}} \right)dx $$
1
vote
0answers
18 views

Finding an anti-derivable non-linear function of a Fourier partial sum

I'm working on a project where I need to compute definite integrals of the composition $\sigma(g(x))$, where $\sigma(x)$ is any non-linear amplifier/activation function and $g(x)$ is the sum of many ...
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\\ ...
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)} ...
0
votes
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 ...
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 $$\large\lim_{A\to\infty}\frac{1}{A}\int_1^A A^\frac{1}{x}\,\mathrm dx$$ ...
2
votes
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 ...
1
vote
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
votes
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 ...
31
votes
2answers
530 views
+500

Closed form for $\int_0^1\frac{x^{5/6}}{(1-x)^{1/6}\,(1+2\,x)^{4/3}}\log\left(\frac{1+2x}{x\,(1-x)}\right)\,dx$

I need to evaluate this integral: $$Q=\int_0^1\frac{x^{5/6}}{(1-x)^{1/6}\,(1+2\,x)^{4/3}}\log\left(\frac{1+2x}{x\,(1-x)}\right)\,dx.$$ I tried it in Mathematica, but it was not able to find a closed ...
13
votes
2answers
165 views

Integral $\int_0^\infty\text{Li}_2\left(e^{-\pi x}\right)\arctan x\,dx$

Please help me to evaluate this integral in a closed form: $$I=\int_0^\infty\text{Li}_2\left(e^{-\pi x}\right)\arctan x\,dx$$ Using integration by parts I found that it could be expressed through ...
0
votes
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, ...
9
votes
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
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 ...
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 ...
1
vote
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 ? ...
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
61 views

Integration by parts prove integral of $\cos^n x dx$ [closed]

I'm having a problem with one of my questions. How can I prove that $$\int\cos^n x dx=\sin x\cdot\cos^{n-1}x+(n-1)\int\sin^2x\cos^{n-2}x dx?$$
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 ...
4
votes
1answer
342 views

Solving Riemann-Stieltjes integral

I'm having trouble solving this Riemann-Stieltjes integral: $$\int_{- \pi/4}^{\pi/4} f(x)dg(x)$$ where , $f(x):= \begin{cases} \frac{\sin^4x}{\cos^2x}{} &\text{if }x\ge0, \\{}\\ \frac1{\cos^3x} ...
0
votes
0answers
36 views

Yet another asymptotic series that needs to be analyticaly extended

Let $A>0$ and $1\le \mu \le 2$. Consider a following definite integral: \begin{equation} {\mathcal I}(A,\mu) := Re\left[\int\limits_0^\infty e^{-(k A)^\mu}\frac{\left(\gamma+\Gamma(0,\imath ...
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 ...
-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
vote
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
134 views

A necessary condition to $F'(x)=f(x)$ for a continuous function $f$

Theorem: Consider , $$F(x)=\int_a^xf(t)\,dt$$ If the function $f:[a,b]\to \mathbb R$ is continuous then , $F(x)$ is differentiable and $F'(x)=f(x).$ I know that the continuity ...
1
vote
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 ...
13
votes
2answers
435 views

A couple of definite integrals related to Stieltjes constants

In a (great) paper "A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations" by Iaroslav V. Blagouchine, the following ...
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\ ...
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? ...
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 ...
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} ...