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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2
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2answers
26 views

Area of the polar figure enclosed by the circle $r=2$ and the cardioid $r=2(1+cos θ)$

This is exercise 7, of the book Engineering Mathematics by Stroud, Chapter 24, Further Problems section. Here's a graph i made of the figure as i see it: It gives the answer as $π+8$. The integral ...
9
votes
3answers
272 views

A limit in a Feynman “proof” about Fermat's Theorem.

As perhaps some of you already know, Richard P. Feynman, the famous physicist tried a non-orthodox (in his usual way, I suppose) proof of the Fermat's Last Theorem. He tried a probabilistic "proof" ...
1
vote
2answers
27 views

On an inequality for harmonic functions

I'm trying to understand the following; $| u(z)|^{p}= |\int P_{z}u \ d\lambda |^{p} \le (\int P_{z} |u |^{p} \ d\lambda) (\int P_{z})^{p-1} $ where $P_z$ is the Poisson kernel and $u$ is harmonic in ...
0
votes
0answers
19 views

Proving that an integral of several cdf and pdf functions is increasing in a certain parameter.

Basic assumptions: $n\geq3$, $a\leq b\leq c$, $b$ is simply a dummy variable of integration, and $\rho\geq0$. $F(z)$ and $f(z)$ represent the usual general CDF and PDF (no specified distribution here)....
2
votes
3answers
91 views

How to integrate: $f(x) =\frac{ \cos5 x +5 \cos 3x+ 10\cos x}{\cos 6x+6\cos 4x+15 \cos 2x+10}{dx}$

$$f(x) =\frac{ \cos5 x +5 \cos 3x+ 10\cos x}{\cos 6x+6 \cos 4x+15 \cos 2x+10}{dx}$$ I applied Eulear formula $$e^{{ \iota}x} = u$$ $$\frac{ u^5 +\frac{1}{u^5}+ 5 (u^3+ \frac{1}{u^3}) + 10(u+ \frac{1}...
15
votes
1answer
173 views

Family of definite integrals involving Dedekind eta function of a complex argument, $\int_0^{\infty} \eta^k(ix)dx$

The Dedekind eta function is denoted by $\eta(\tau)$, and is defined on the upper half-plane ($\Im \tau >0$). Put $\tau = i x$ where $x$ is a positive real number. The function has the following ...
1
vote
0answers
32 views

Fourier Series and Coefficient Calculation Under two minutes

Example of one Question for preparing the entrance exam: Fourier series of function: $$ f(x)=f(x+2\pi), f(x) =\left\{ \begin{array}{rcr} 1 & & -\pi <x<0 \\ \sin x & &...
1
vote
2answers
68 views

Closed-form Solution for series involving incomplete Gamma Function

I am working on a solution for an intgeral that leads to a series that I am stuck at. Below is what I have done and how I got to the final series. Any ideas on how to solve the series at the end? \...
0
votes
0answers
27 views

Approximation of series using integral

In notes of statistical physics I found the following approximation $$\sum\limits_{n=0}^{\infty}F\left(n+\frac{1}{2}\right)\approx \int_{0}^{\infty}F(x)dx+\frac{1}{24}F'(0)$$ for $F$ such that the ...
52
votes
6answers
5k views

Is there a function whose antiderivative can be found but whose derivative cannot?

Does a function, $f(x)$, exist such that $\int f(x) dx $ can be found but $f' (x)$ cannot be found in terms of elementary functions. For example, if $f(x)=e^{x^2}$, then the derivative is easily ...
0
votes
1answer
18 views

How small need it be to approximate integral as one area of product of initial value times length.

$$\left(\int_{t}^{t+\Delta t}a(t')dt'\right), a(t) \text{ is scalar}$$ How small need $\Delta t$ be to approximate $$\left(\int_{t}^{t+\Delta t}a(t')dt'\right)$$ as $$a(t)\Delta t$$ [ Just one ...
31
votes
3answers
659 views

Triple Euler sum result $\sum_{k\geq 1}\frac{H_k^{(2)}H_k }{k^2}=\zeta(2)\zeta(3)+\zeta(5)$

In the following thread I arrived at the following result $$\sum_{k\geq 1}\frac{H_k^{(2)}H_k }{k^2}=\zeta(2)\zeta(3)+\zeta(5)$$ Defining $$H_k^{(p)}=\sum_{n=1}^k \frac{1}{n^p},\,\,\, H_k^{(1)}\...
2
votes
1answer
71 views

Integrating factor

Can anyone give me some hints as to how to solve the following question? I have to show that the equation below has an integrating factor of the form $t^2\theta^c$ where $c$ is an integer. $\...
-1
votes
0answers
16 views

Calculate integral $x_1^{k-1}x_2^{l-k-1}(1-x_1-x_2)^{n-l}$ [on hold]

Calculate $\int_0^{1-x2}\ x_1^{k-1}x_2^{l-k-1}(1-x_1-x_2)^{n-l}dx_1$ If $n,l,k$ is fixed -it's easy - just expand polynomial and cause similar terms. But what I can do in this common case?
2
votes
0answers
12 views

Integration over the Haar measure of a compact Lie group preserves smoothness?

Let $G$ be a compact Lie group. Then there is a unique Haar (probability) measure on $G$. If $f \colon G \to \mathbb{R}$ is a smooth function, is the function $$ G \to \mathbb{R}, \qquad x \mapsto \...
5
votes
5answers
266 views

How to integrate $\int \frac{1}{\cos(x)}\,\mathrm dx$

could you help me on this integral ? $$\int \frac{1}{\cos(x)}\,\mathrm dx$$ Here's what I've started : $$\int \frac{1}{\cos(x)}\,\mathrm dx = \int \frac{\cos(x)}{\cos(x)^2}\,\mathrm dx = \int \frac{...
4
votes
1answer
97 views

How to integrate $\displaystyle\int e^{\frac{x^3}{3}} \mathrm{d}x$

While solving $y'=x^2-e^y$ I'm stuck on the last step that requires to evaluate this integral. $$\displaystyle\int e^{\frac{x^3}{3}} \mathrm{d}x$$ I don't know how to approach it. I know that it ...
0
votes
1answer
65 views

Integrate $\int\frac{2x+6}{x^2+6x+18}\mathrm dx$

My problem is: $$\int\frac{2x+6}{x^2+6x+18}\mathrm dx$$ This is part of a larger integral so I didn't see the $y=x^2+6x+18$ substitution straight away. My concern is what was wrong with my working in ...
2
votes
1answer
136 views

Laplace Challenge in One Examples, Is there any help?

this question is taken from 2014 exam on CE Entrance Exam, Question $32$ on the end of page $6$. Consider the Laplace equation of following polar coordination, $$\frac{1}{r}\frac{\partial}{\partial ...
0
votes
2answers
42 views

Volumes by cylindrical shells

Find the volume bound by $$xy=1;x=0;y=1;y=3; $$rotated about $x$-axis. In my attempt, I determined that the shell height will be $1/y$ and the shell radius will be $y-1$ Clearly these are wrong as ...
0
votes
1answer
16 views

What is the area between two first order taylor series approximations as they become closer to eachother

Let's say that $y=\sin{x}$. Then the first order taylor series approximation about $c$ is $g(x)=\sin{(c)}+\cos{(c)}(x-c)$. Note that this is also equivalent to the line tangent to the curve $\sin{x}$ ...
-2
votes
3answers
108 views

$\iiint xyz~ dx~dy~dz$ over positive octant sphere $x^2+y^2+z^2= 4$? Please help me about about question solution [on hold]

$\iiint xyz~ dx~dy~dz$ over positive octant sphere $x^2+y^2+z^2= 4$? I have problems figuring out the limits. Complete answer will be appreciated. Please help me about about question solution
0
votes
0answers
41 views

Showing propagation in ordinary differential equation

I have a very simple linear First Order Homogeneous Differential Equation with time but this makes me ponder upon the definition of derivative and integral. $$\frac{dy}{dt} = -H(t)y$$ where $y, H(t)$ ...
1
vote
3answers
35 views

Deriving the volume of a sphere through integation

Disclaimer: I'm basing my information off of this link in the case where I do not provide enough context. All calculus I've learned has been through curiosity, digging up old textbooks and google (I ...
1
vote
0answers
28 views

Integral of a summation related to $\sin$ expansion

I am trying to evaluate the following integral. It has similarity to the Maclaurin expansion for $\sin$. $$\int_{-\infty}^\infty{\sum_{n=0}^{\infty}\frac{(-1)^n}{\left(2n+x^2\right)!}}\text{dx}$$ ...
2
votes
3answers
70 views
0
votes
2answers
320 views

Complex form of Fourier Series

So, the last part of the university syllabus in the chapter of Fourier Series is: ...
3
votes
1answer
27 views

Algebra $A$ and its Gelfand spectrum

Let $A$ be the set of all function $f$ on $\mathbb{R}$ of the form $$ f(x)=d+\int\limits_{0}^{\infty}e^{ixt}k(t)dt,\qquad\quad x\in\mathbb{R}, $$ where $d\in\mathbb{C}$ and $k\in L_1([0,\infty])$. The ...
0
votes
2answers
613 views

Any tips to solve this integral : $I_1 = \int \ln(x^2)e^{\sin(x)}\sin(x^{\cos(x)}) dx$

Background: I was making new expressions to see whether I could efficiently find their derivatives... After having done that, I've started trying to integrate most of them; obviously most of them don'...
-5
votes
1answer
42 views

Need help with an equation, please. [on hold]

Please help. Don't know how to solve it :( $$\begin{align*} J=&\int_{AB}xy\,ds\\\\ AB:\;&\begin{cases}x=\cos t\\y=\sin t\end{cases}\qquad t\in[0,\pi/2]\quad A=(1;0)\quad B=(0;1) \end{align*...
4
votes
3answers
157 views

Derivative of a negative order?

Below, $\Delta$ means taking the derivative, $\frac{d}{dx}$. For $n\in\mathbb{Z}$, $n\geq 0$, we have $$\Delta^n\sin{x}=\sin{(x+n\tau/4)} \\ \Delta^n\cos{x}=\cos{(x+n\tau/4)}$$ I found that out while ...
5
votes
5answers
8k views

Does there exist a function that is differentiable but not integrable? or integrable but not differentiable?

It has become very complicated to me to find out a function which is differentiable but not integrable or integrable but not differentiable.
0
votes
1answer
28 views

Integration by parts vs expanding giving different answers.

You are given the probability density function of a random variable X. $$f_X(x) = 2x$$ $$0<x<1$$ Find the difference between the third central moment and the second central moment of this ...
0
votes
1answer
54 views

Can I show that $\int_{\gamma(0;r)} \frac{1}{z-a} dz = 0$ when $|a|>r>0$ without using Cauchy Theorem?

I encountered this problem as a previous result of an exercise in a text book way before proving Cauchy Theorem, so I think there must be another way to prove it without it. Show that $\int_{\...
18
votes
1answer
415 views

How to find $\lim_{n \to \infty} \int_0^1 \cdots \int_0^1 \sqrt{x_1+\sqrt{x_2+\sqrt{\dots+\sqrt{x_n}}}}dx_1 dx_2\dots dx_n$

Here I mean the limit of the following sequence: $$p_1=\int_0^1 \sqrt{x} ~dx=\frac{2}{3}$$ $$p_2=\int_0^1 \int_0^1 \sqrt{x+\sqrt{y}} ~dxdy=\frac{8}{35}(4 \sqrt{2}-1) = 1.06442\dots$$ $$p_3=\int_0^...
2
votes
2answers
49 views

Limit and convergence/divergence of an integral

I was working on a problem concerning the function $$f(x) = \frac{x^2}{\ln(x)^\sqrt{x}}$$ asking for the value of $$\lim_{x \to \infty}f(x)$$ and for the convergence/divergence of $$\int_2^\infty f(x) ...
0
votes
1answer
23 views

Bounds on flux integrals

What are some handy upper bounds for surface integrals (and their proofs)? Specifically, suppose $f$ is a bounded function on a surface $S$. Do we have $$ \int_{\partial S} F \cdot n \; \mathrm{d}S \...
4
votes
3answers
387 views

Integration by parts: is this legitimate way of using?

Is it legitimate to write $$\int_0^a\mathrm{d}x\,f(x)g(x)=\left[f(x)\int_0^x\mathrm{d}x\,g(x)\right]_0^a-\int_0^a\mathrm{d}x\,\frac{\mathrm{d}f(x)}{\mathrm{d}x}\int_0^x\mathrm{d}x\,g(x)$$ Thanks.
0
votes
0answers
30 views
+50

Clarify and justify how get the derivative of the Laplace transform of the Buchstab function

I would like to justify that the derivative with respect to $s$ of the Laplace transform of the Buchstab function is $$\int_1^\infty u\omega(u)e^{-su}du=\frac{e^{-s}}{s}\exp\left(\int_0^\infty \frac{e^...
2
votes
1answer
63 views

Integrating $e^{a\cos(\phi_1-\phi_2)+b\cos(\phi_1-\phi_3)+c\cos(\phi_2-\phi_3)}$ over $[0,2\pi]^3$

I am trying to integrate the following function. (it arises in channel modeling in wireless communications, Rayleigh random variables)..Any help is appreciated.Thanks $$\int_0^{2\pi}\int_0^{2\pi}\...
-11
votes
1answer
174 views

What is $\int \frac{\mathrm{d} f(t)}{\int f(t) \, \mathrm{d} t}\, \mathrm{d} t$?

The integral $\int \frac{\mathrm{d} x}{\mathrm{d} y}^{-1} \, \mathrm{d} x$ is $ y + c$ subject to some interesting qualifications about continuity and holes. I think I can evaluate $\int \frac{\...
1
vote
1answer
18 views

Area between two polar curves method

The question is not too hard. I sketched them and they were correct which was not too bad. I then did the second part by finding the intersection points between the two curves which are $\frac{\pi}{...
1
vote
0answers
26 views
+50

Mean continuity of gradient

Let $f:\mathbb R^n\longrightarrow R$ be a differentiable function, and suppose $\nabla f$ is bounded. Prove that $$\lim_{r\to 0}\frac{1}{\omega_n r^n}\int_{B_r(x)}[\nabla f(y)-\nabla f(x)] dy=\...
0
votes
1answer
31 views

N-order differential equations

Suppose that we have n-order differential equation like $$h(x)=?$$ Is it possible to find a general solution for all n? $$(x^n+1).|h'(x)|^n=const.$$.
0
votes
2answers
63 views

Maxwellian integral : is there a closed form?

$f_A(x,y)=\int_0^\infty du \frac{u \left(e^{-\frac{(u-x)^2}{2 A}}-e^{-\frac{(u+x)^2}{2 A}} \right)}{\sqrt{2 \pi } \sqrt{A} x \left(y^2+u^2\right)} $ is there a closed form? I was able to find ...
-2
votes
1answer
38 views

Integrating differential forms on curves [closed]

How can I integrate the differential form $$\omega=x\,dx+y\,dy+z\,dz$$ in $\mathbb R^3$ on the curve $$c:[0,2\pi]\to\mathbb R^3: t\mapsto (e^{t\sin t}, t^2-2\pi t, \cos \frac{t}{2})?$$ Some advice ...
1
vote
1answer
43 views

Equivalent of $\int_2^{+ \infty} e^{\Gamma (t) \log x}dt$ when $x \to 1$

I wonder if the equivalent : $$ \int_2^{+ \infty} e^{\Gamma (t) \log x}dt $$ for $x \to 1^{-}$ (i.e the first term in the asymptotic expansion) had been studied ? Is it tricky to get an equivalent ?...
3
votes
1answer
93 views

Integrate the following: $\int \frac {\log x}{\sqrt {1-x^2}}dx$

What I tried : $\int \frac {\log x}{\sqrt {1-x^2}} dx$ = $\log x \int \frac {1}{\sqrt {1-x^2}} dx$ - $\int \frac{1}{x} (\int \frac {1}{\sqrt {1-x^2}} dx) dx$ Now, $\int \frac {1}{\sqrt {1-x^2}} ...
0
votes
1answer
28 views

Integrating Over a Product of (Non-Separable) Piecewise Functions (Hyper-Solid Angle of a Convex Polyhedral Cone)

My problem is as follows: given a function $f:\mathbb{R}^n \rightarrow \mathbb{R}$ where $n$ is some integer of order 10 and $f$ is defined by a product of (non-separable) linear piecewise functions, ...
2
votes
0answers
28 views

Integral of familly of curves

Let $f_n(t):=t^{n+m+1/3}e^{-(t^{-n}+t^{-m}+t^2)}$, where $n,m\geq 1$. I have been having difficulty calculating this ingetral on $\mathbb{R}$. Please help, thanks.