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 views

Evaluating an integral (real part, cauchy pdf)

I was trying to find a characteristic function for a cauchy pdf, without going into contour integration (because I have no idea how to do it). I have: ...
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0answers
5 views

Surface Integral of a harmonic function and mean value property

I want to find a general expression of the following integral, where $h$ is a harmonic function (we're in $\mathbb{R}^2$): $\int_{|x-y|\leq a^2}\frac{h(y)}{\sqrt{a^2-|x-y|^2}}dy$ I think I can ...
15
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5answers
551 views

Is there any integral for the Golden Ratio?

This is a curiosity. I was wondering about math important/famous constants, like $e$, $\pi$, $\gamma$ and obviously $\phi$. The first three ones are really well known, and there are lots of integrals ...
0
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1answer
14 views

$g(x) \ge 0$ Riemann integrable on $[a,b]$ then for each subinterval $\int^b_a g(x)dx \ge \int^d_c g(x)dx$

Let $g(x) \ge 0$ Riemann integrable on $[a,b]$. Show using Riemann sums that for each subinterval $[c,d] \subset [a,b]$: $$\int^b_a g(x)dx \ge \int^d_c g(x)dx$$ I thought that we should ...
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0answers
12 views

integration of greatest integer function

What the integral of $$\int [cot(x)]dx$$ over $0-\pi$ where $[.]$ represents greatest integer function i know integral of cot is $|\log(sin(x))|$ but log isnt defined for $0$ or is there something ...
7
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1answer
98 views

Closed form for Euler sum with $H_{2n}$?.

I ran across this Euler sum while trying to evaluate an integral. I mentioned it in another thread, but though perhaps asking about it separate may be a good idea. Is there a closed form for this ...
0
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1answer
35 views

Definite integral with cube roots of trig functions [on hold]

Find a closed form for the following integral: $$\int _{\pi/6} ^{\pi /3} \frac {\sqrt[3]{\sin x}}{\sqrt [3]{\sin x} + \sqrt[3]{\cos x}}dx$$ I think the answer is found out using some properties. So ...
1
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1answer
17 views

Computation of an integral depending on the Legendre polynomials

Let $P_l$ be a Legendre polynomial ($l$ is an integer). I want to know why the quantity $$ v_l(k):=(-i)^l\int_{-1}^{+1}\mathrm{e}^{ikx}\,P_l(x)\;\mathrm{d}x $$ is real?
1
vote
1answer
47 views

Flux Through a Closed Curve - Orientation

I want to compute $$\int_{C}\boldsymbol{F}\cdot\boldsymbol{n}\, ds\qquad\quad \boldsymbol{F}=\langle x, y^2\rangle$$ where $C$ is the curve given by the triangle with vertices $(-1,0)$, $(0,1)$ and ...
2
votes
1answer
21 views

Change of variables, integration

In a finite element analysis, I am evaluating the following integral: $$\int_{0}^{h}\left ( 1-\frac{x}{h} \right )*\left ( x \right )dx$$ but I want to apply a transformation from x to integrate ...
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0answers
13 views

Construction of the Area Function

I am following calculus by Tom M Apostol in which he has given the Axiomatic definition of the Area Function We assume there exists a class M of measurable sets in the plane and a set function a, ...
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1answer
33 views

Evaluating this surface integral

I want to evaluate the following surface integral $$\int_S d S \, \, \exp\left(\frac{\Gamma^2 (x^2 \sigma_x^2 + y^2 \sigma_y^2 + z^2 \sigma_z^2) - 2 c R \Gamma (x x_0 + y y_0 + z z_0)}{2 c^2 ...
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votes
1answer
44 views

How to find the center of mass in this problem

How can I find the centre of mass of the surface of the sphere $x^2+y^2+z^2=a^2$ that is contained in the cone $z\tan(\gamma)=\sqrt{x^2+y^2}$, $0 \lt \gamma \lt$ $\pi/2$ a constant, where the density ...
4
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1answer
58 views

“Increasingify” a function

Let $f : [a,b] \rightarrow \mathbb{R}$ be a $C^1$ function such that $f$ is monotonic on each $[t_k, t_{k+1}]$, with $a = t_0 < t_1 < ... < t_N = b$. Let g be the increasing-ified version of ...
2
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0answers
77 views

Prove that the following function is convex?

I am trying to prove that the function $$g(\alpha)=\ln\Big(\sum_{i=1}^{n}t_i^\alpha+A(\alpha)\Big) ~~t_i, \alpha>0,$$ where $A(\alpha)=\int_{a}^{b}x^{\alpha+\beta-1}e^{-\lambda x^\beta}\,dx$,is ...
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2answers
27 views

Find the original function by using convolution theorem

Seems like I don't know how to apply convolution theorem on this problem properly, I would appreciate some help and a brief explanation how did you solve it if you do it. ...
0
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1answer
32 views

If $f(x)=\lim_{n\to\infty}n^2(e^{\frac{k}{n}\ln\sin x}-e^{\frac{k}{n+1}\ln\sin x})$ where $0<x<\pi$, $n\in\mathbb{N}$

If $f(x)=\lim_{n\to\infty}n^2(e^{\frac{k}{n}\ln\sin x}-e^{\frac{k}{n+1}\ln\sin x})$ where $0<x<\pi$, $n\in\mathbb{N}$ and $\int_0^{\frac{\pi}{2}}f(x)dx=-\frac{\pi}{k}\ln4$, then the value of ...
-1
votes
1answer
36 views

Help with vector triple integral problem

Prove that $$\iiint_{D}(\vec a \cdot \vec R)(\vec b \cdot\vec R)(\vec c \cdot\vec R) \,dx\,dy\,dz=\frac{(\alpha\beta\gamma)^2}{8r}$$ Where the $\vec a , \vec b,\vec c$ are constant vectors, $\vec ...
1
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1answer
40 views

Integrating inverse trig function with radicals

$$\dfrac{x + 5}{\sqrt{9-(x-3)^2}}$$ It's a inverse trig integration problem. I tried to separate the numerators but made my problem worse. Any advice?
91
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4answers
4k views

Is there an integral that proves $\pi > 333/106$?

The following integral, $$ \int_0^1 \frac{x^4(1-x)^4}{x^2 + 1} \mathrm{d}x = \frac{22}{7} - \pi $$ is clearly positive, which proves that $\pi < 22/7$. Is there a similar integral which proves ...
2
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2answers
58 views

Integral $\frac{\sin(x)}{x}$ finite domain

I have seen a question asking to find the value of $\int_{-100}^{100} \frac{\sin{x}}{x} dx$. I have to confess that I didn't think this was possible. If I expand the $\sin$ using Taylor series, ...
6
votes
2answers
206 views

Closed form for a zeta series :$\sum^\infty_{k=2}\frac{(-1)^{k-1}\zeta(k)}{(k+2)2^{k+2}}$

It is not that diffcult to derive \begin{align} \sum^\infty_{k=2}\frac{(-1)^{k-1}\zeta(k)}{k2^k}=&-\frac{\gamma}{2}+\ln\left(\frac{2}{\sqrt{\pi}}\right)\tag{1}\\ ...
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0answers
91 views
0
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3answers
75 views

Integrate $\int{\frac{1}{(x+1)(x+2)^2(x+3)^3}dx}$

How to integrate this $$\int{\frac{1}{(x+1)(x+2)^2(x+3)^3}dx}$$ I tried to use that $$\int{\frac{1}{(x+1)(x+2)^2(x+3)^3}dx} = P_{1}(x)/Q_{1}(x) + \int{P_{2}(x)/Q_{2}(x)dx}$$ where ...
0
votes
0answers
63 views

Solving an indefinite integral problem [on hold]

The given problem is $$ \int \frac{2^{\sin x}}{2^{\sin x} + 2^{\cos x}} dx$$ please help me solving this indefinite integral problem...thank you very much, actually I have solved an definite integral ...
2
votes
2answers
79 views

Complicated surface integral/line integral.

Problem Compute the integrals $$I=\iint_\Sigma \nabla\times\mathbf F\cdot d\,\bf\Sigma$$ And $$J=\oint_{\partial\Sigma}\mathbf F\cdot d\bf r$$ For $F=(x^2y,3x^3z,yz^3)$, and ...
0
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0answers
15 views

Change of variables from unit unit ball to another ball for integration

What is the general formula for changing coordinates for integration from the unit ball to another ball? For example, if I wanted to change from integrating $f(x-r)$ over $B(0,1)$, the open ball about ...
0
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1answer
15 views

Trapezoid rule for finding coefficient

If we know that $\int_{a}^b t(x)=h \sum_{k=1}^2 dk * t(a+kh)+O(h^m)$ where $h=\frac{b-a}{3}$, how do we find the coefficient d1, d2 and m in the equation? Answer says that d1=3/2, d2=3/2, m=3 I ...
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2answers
89 views

integrate $\int \frac{dx}{x\sqrt{9+4x^2}}$

$$\int \frac{dx}{x\sqrt{9+4x^2}} $$ I understand I need to use $x=\frac{3}{2}\tan\theta$ trigonometric substitution So I got to: $$\int {\frac{3}{2\cos^2\theta}\over \frac{3}{2} \tan\theta ...
1
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0answers
35 views

Indefinite integration and definite integration calculation [on hold]

Given the functions: $$\begin{cases} A(x)= \dfrac {x\ln(1+\ln(x))}{1+x^{4/3}}\\[2ex] B(x)=A(2x-1)-A(2x) \end{cases}$$ for $b>1$ find out: $$\int_{b}^\infty A(x) dx - \frac12\int_{2b-1}^{2b} B(x) ...
6
votes
2answers
94 views

Anti-derivative of continuous function $\frac{1}{2+\sin x}$

I use tangent half-angle substitution to calculate this indefinite integral: $$ \int \frac{1}{2+\sin x}\,dx = \frac{2}{\sqrt{3}}\tan^{-1}\frac{2\tan \frac{x}{2}+1}{\sqrt{3}}+\text{constant}. $$ ...
2
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1answer
47 views

Finding Fourier series constant and integral

I have been studying Griffith's Intro to Electrodynamics. I am studying differential equations and Fourier series. I am studying the problem discussed here: Why is this allowed? ("Fourier's ...
0
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2answers
27 views

Integral of a measurable function

I do not know what should i keep as title for this question... Question goes like this.. Let $f:\mathbb{R}\rightarrow [0,\infty)$ be a measurable function. If $\int_{-\infty}^{\infty}f(x)dx=1$ prove ...
14
votes
1answer
316 views

Mathematical meaning of certain integrals in physics

While studying on texts of physics I notice that differentiation under the integral sign is usually introduced without any comment on the conditions permitting to do so. In that case, I take care of ...
2
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0answers
55 views

difficult integral $\int_0^{\pi/2}\frac{x^2({1+\tan^2 x})^2}{\sqrt{\tan x}({1-\tan x})}\sin{4x}dx$

This is a complicated integral, the numerical value appears to me correct.Therefore how to prove this result?$$I=\int_0^{\pi/2}\frac{x^2({1+\tan^2 x})^2}{\sqrt{\tan x}({1-\tan ...
0
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0answers
33 views

Proving an inequality involving integrals?

I am trying to prove that $$[\sum_{i=1}^{n}(\ln t_i)^2 t_i^\alpha+A^{\prime \prime}(\alpha)][\sum_{i=1}^{n}t_i^\alpha+A(\alpha)]\ge[\sum_{i=1}^{n}(\ln t_i) t_i^\alpha+A^{\prime}(\alpha)]^2$$ where ...
1
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1answer
173 views
+50

How to solve integrals of type $ \int\frac{1}{(a+b\sin x)^4}dx$ and $\int\frac{1}{(a+b\cos x)^4}dx$

$$\displaystyle \int\frac{1}{(a+b\sin x)^4}dx,~~~~\text{and}~~~~\displaystyle \int\frac{1}{(a+b\cos x)^4}dx,$$ although i have tried using Trg. substution. but nothing get
11
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3answers
288 views

How to compute $\int_0^\infty \frac{x^4}{(x^4+ x^2 +1)^3} dx =\frac{\pi}{48\sqrt{3}}$?

$$\int_0^\infty \frac{x^4}{(x^4+ x^2 +1)^3} dx =\frac{\pi}{48\sqrt{3}}$$ I have difficulty to evaluating above integrals. First I try the substitution $x^4 =t$ or $x^4 +x^2+1 =t$ but it makes ...
0
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0answers
49 views

how to evalute this equality

I want to prove this equality $$ \frac{1}{2\pi}\frac{(x-y)\cdot y}{(x_1-y_1)^2+(x_2-y_2)^2}= \frac{ab}{4\pi}\frac{1}{a^2\sin^2(\alpha+\beta)+b^2\cos^2(\alpha+\beta)}.\tag{1}$$ where ...
1
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2answers
42 views

Sign of the error in Simpson's rule

Let $f : [a,b] \to \mathbb{R}$ be a $C^\infty$ function. The Riemann integral $I = \int_a^b f(x)\,dx$ can be approximated by using Simpson's rule: $$I \approx S = \frac{b-a}{6} \left[ f(a) + 4 ...
6
votes
3answers
1k views

How to integrate $\int_0^1x^a(1-x)^bdx$

I have a question about an equation I am trying to integrate, the integral is: $$\int_0^1 x^a (1 - x)^b ~dx,$$ where $a, b > 0$. Any assistance with this problem would be appreciated.
0
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1answer
21 views
21
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2answers
692 views

Integral: $\int_{-\infty}^{\infty} \frac{dx}{(e^x+x+1)^2+\pi^2}$

I am looking for real analytic methods to prove the following: $$\int_{-\infty}^{\infty} \frac{dx}{(e^x+x+1)^2+\pi^2}=\frac{2}{3}$$ I have seen a similar problem on the website but if I remember ...
1
vote
1answer
26 views

Proving a reduction formula. $\cos^n (2x)$

Establish a reduction formula for $$\int \cos^n (2x)dx$$ My attempt, Let $I_n=\int \cos^n 2x dx$ $=\int \cos^{n-1}2x (\cos 2x dx)$ Let$$u=\cos^{n-1}2x$$ $$du=-2(n-1)\cos^{n-2}2x (\sin 2x)dx$$ ...
0
votes
1answer
25 views

How to Proceed in Solving this Equation

Let $f: [0,\infty)\to \mathbb{R}$ a non-decreasing function. Then show this inequality holds for all $x,y,z$ such that $0\le x<y<z$. \begin{align*} & (z-x)\int_{y}^{z}f(u)\,\mathrm{du}\ge ...
0
votes
1answer
485 views

Taylor expansion, integration by parts, and the integration of dt.

So my notes say, for a continuous function we have $$ \int_a^x f'(t)dt = f(x) - f(a) \tag 1 $$ which I understand. So re-arranging gives. $$ f(x) = f(a) + \int_a^x f'(t)dt \tag 2 $$ or $$ f(x) ...
46
votes
12answers
1k views

Why is it not true that $\int_0^{\pi} \sin(x)\; dx = 0$?

I know the following is not right, but what is the problem. So we want to calculate $$ \int_0^{\pi} \sin(x) \; dx $$ If one does a substitution $u = \sin(x)$, then one gets $$ \int_{\sin(0) = ...
0
votes
0answers
27 views

Bounding an integral

I'm trying to show that the following integral ( a solution for the non-homogeneous transport equation ) has this bound: $$ \begin{equation*} \left\|{ \int_{0}^{t} f(x+b(w-t),w) dw ...
0
votes
1answer
25 views

Surface are of a curve $y=\sin \left(\frac{\pi x}{6} \right)$ rotated about the $x$ axis.

I'm doing a problem involving finding the surface area of the curve for $y=\sin \left(\frac{\pi x}{6} \right)$, rotated about the $x$ axis, for $[0 < x < 6]$. I got as far as $\frac{72}{\pi} ...
4
votes
0answers
45 views

An alternative way to determine when $\int_{0}^{\infty} \cos(\alpha x) \prod_{m=1}^{n} J_{0}(\beta_{m} x) \, dx =0$

Let $J_{0}(z)$ be the Bessel function of the first kind of order zero, and assume that $\alpha$ and $\beta_{m}$ are positive real parameters. When $|z|$ is large in magnitude and $-\pi < \arg(z) ...