Questions tagged [integration]

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 the integrals being considered. This tag often goes along with the (calculus) tag.

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0answers
14 views

Showing time it takes a particle to reach a point is finite

This question comes from a physics text, but I believe my issue is a mathematical one. Suppose a particle is moving along the real line starting at $x_0$ subject to the potential $V$ with $V(x)<...
3
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0answers
57 views

Does this $\int_{0}^{1}\left(\sqrt{1-x^2}^{\sqrt{1-x^2}^{\cdots}}\right) dx$ converge?

I tried to evaluate $\int_{0}^{1}\left(\sqrt{1-x^2}^{\sqrt{1-x^2}^{\cdots}}\right) dx$ , using the fact $x= \cos t$ yield to have integrand using $\sin $ function, for one iteration by means $\int_{...
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1answer
12 views

How do I integrate a partial differential equation with a constant quantity?

If I have a partial differential equation of the form: $$\left( \frac{\partial y}{\partial x}\right)_{p/x} = -\frac{k \hspace{1mm}z(p/x)}{x} + \frac{ky}{x}$$ where the $p/x$ subscript represents that ...
3
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0answers
74 views

Evaluate $\int_{-\infty}^0 \big(\log(1-e^{x^3})\big)^{1/3}~dx$

How can I evaluate $$\int_{-\infty}^0 \left(\log \left(1-e^{x^3} \right)\right)^{1/3}~dx$$ Thank you. I tried u-sub, int. by parts, other classical methods, and partial fractions?
3
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1answer
64 views

Does this $\int_{0}^{\infty}(\frac{\log x}{e^x})^n dx$ always have a closed form for $n$ being positive integer ? what about its irrationality?

It is known that $\int_{0}^{\infty}\left(\frac{\log x}{e^x}\right)^n dx=-\gamma$ for $n=1$ and for $n=2$ we have :$\frac{1}{12}(\pi^2+6(\gamma+\log 2)^2)$ and for $n=3$ we have this form , What I have ...
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0answers
14 views

Integration of nested function using exponential inside trigonometric

I've been working with a chirp function, as follows: $$y=A\sin\left[2\pi f_0\left(\frac{k^t -1}{\ln(k)}\right)\right]$$ where A is the amplitude and is constant over the whole frequency range. $f_0$ ...
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0answers
49 views

Calculation of $\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}\left\{\frac{y}{x}\right\}\left\{\frac{z}{y}\right\}\left\{ \frac{x}{z}\right\}\,dx\,dy\,dz$ [duplicate]

I 'd like to calculate the integral of the following expression in cube $$\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}\left\{\frac{y}{x}\right\}\left\{\frac{z}{y}\right\}\left\{ \frac{x}{z}\right\}\,dx\,dy\,...
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47 views

Does a closed solution of this integral exist

I saw a problem on Facebook with this integral except it was a definite integral from -2 to +2 and the answer was 0 since the function was odd. I am wondering if a closed solution exists or if this ...
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1answer
11 views

Proof inner-product on square-integrable products is always well defined?

The inner product of any two square-integrable functions $f$, $g$ is defined to be $\int f^*(x) g(x) dx$ (forgive my ugly maths layout). But how do we know this is never ill-defined? Why does the ...
0
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0answers
13 views

Sum of two independent Rayleigh variables - find value for specified of probability of exceedance

I have the following equation for the convolution of two Rayleigh distributions (X,Y) based on significant values s1 and s2: ...
0
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1answer
65 views

Show that $\int_{0}^x \frac{1}{1+t^4} dt = x - x^5/5 + x^9/9 … $ where $\lvert x\rvert$ < 1

Show that $\int_{0}^x \frac{1}{1+t^4} dt = x - x^5/5 + x^9/9 ..... $ where $\lvert x\rvert$ < 1 I tried expanding using binomial theorem, but I am unable to prove the series and the integral will ...
2
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2answers
53 views

Integration of an improper integral

I have an integral that has two individually divergent parts. Wolfram says that the answer is \begin{equation}\frac{\ln(s+1)}{s},\end{equation} but I cannot figure out how it's done. The integral is \...
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0answers
10 views

Is this equality about integral on dyadic product true?

Is this equality in the attached picture true? The curl is is done over $r$ not $r_0$, unlike the integral. The symbol $G$ with two pars, is a dyadic.
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0answers
21 views

Almost a variant of coupling inequality

Given a pdf $f$ that is non decreasing in the interval $a \le x \le b$, show that for any $s>0$, $\int_{a}^{b} x^{2s}f(x) dx \ge \frac{b^{2s+1}-a^{2s+1}}{(2s+1)(b-a)}\int_{a}^{b} f(x) dx$ I saw the ...
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1answer
27 views

Calculate line integral $\int_{\ell} y \cos x d \ell$

I am asking to calculate the integral $$\int_{\ell} y \cos x d \ell$$ while $\ell$ is the graph of the function $\phi(x)=sin(x)$ in the domain $x \in [0,\frac{\pi}{2}]$ . So what I understand is that $...
3
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1answer
29 views

The operator norm $\|L\|$

Let $C_0([0, 1])$be a subspace of $C([0, 1])$, a functional space consisting of real-value continuous functions over the interval $[0, 1]$, such that $C_0 ([0, 1]) = \left\{ f \in C([0, 1]) \mid \...
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0answers
8 views

Deriving density of measure from SDE

I'm trying to derive the density of a sub-measure valued process $\nu_t$, which is characterized by the SPDE for some test function $\phi\in\{\phi\text{ is a Schwartz function with }\phi(0)=0\}$ \...
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0answers
49 views

How to solve $f(y)=\int_{0}^{2\pi} e^{p\cos (y) - qy} \,dy$ [closed]

How to solve $f(y)=\int_{0}^{2\pi} e^{p\cos(y)-qy}\, dy$ I think this is probably related to the Bessel function, but I am not sure how to solve it. After a few days of trying, I even suspected that ...
0
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1answer
41 views

Prove that for $b > 0$, $\iint_{\mathbb R^2}e^{-b({x^2}+{y^2})} dA = \frac{\pi}{b}$.

Using polar coordinates, prove that for $b > 0$, $$\iint_{\mathbb R^2}e^{-b({x^2}+{y^2})} dA = \frac{\pi}{b}$$ Using $\frac{\pi}{b}$, also prove that $$\int_{-\infty}^{\infty} e^{-b{x^2}} dx = \...
-5
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1answer
28 views

find error upto four decimal point of the following integral .

Find the value of $$\int_0^{0.8} \frac{\sin(x)}{x} \text{d} x$$ correct to four decimal places. The answer to question is 0.7721..I wanted to know about correct ...
0
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1answer
45 views

Evaluate the Improper Integral(help) [closed]

I encountered the following integral while solving a log-normal distribution question. Initially, I thought since its a odd function, it evaluates to zero. But I think, since its a improper integral, ...
0
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1answer
49 views

Solution of $\int_a^b f(z)g(z)dz=0$

Fix $a\in[0,1]$ and let $b$ an arbitraty value in $[a,1]$. Moreover, there is a positive and continuous function $g:[0,1]\rightarrow(0,\infty)$. I want to find a solution $f$ of $$\int_a^bf(z)g(z)dz=0$...
3
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1answer
27 views

Constructing a locally integrable function

Let $\epsilon\in(0,1)$ and $F^{\epsilon}:\mathbb{R}^2\to\mathbb{R}$ defined by $$F^{\epsilon}(x)=\log(|x|^2+\epsilon^2)$$ How can I construct a $g \in L^1_{loc}({\mathbb{R}}^2)$ such that $$|F^{\...
4
votes
5answers
99 views

How to integrate $\int \sqrt{1-\dfrac{1}{25x^2}}\ dx$?

How to integrate following $$\int \sqrt{1-\dfrac{1}{25x^2}}\ dx$$ What I did is here: $$\int \sqrt{1-\dfrac{1}{25x^2}}\ dx=\int \dfrac{\sqrt{25x^2-1}}{5x}\ dx$$ I substituted $5x=\sec\theta$, $dx=\...
1
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0answers
29 views

Show that the image of $\lambda$ has zero content in $\mathbb R^n$.

Let $\lambda:[0,1] \to \mathbb R^n$ be a class curve $C^1$ Part by part. Then, the imagem of $\lambda$ has null content in $\mathbb R^n$. My attempt, let $\epsilon > 0$, let's consider a ...
3
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4answers
60 views

Find the volume between $z=\sqrt{x^{2}+y^{2}}$ and $x^2+y^2+z^2=2$ in spherical cordinates

I am asking to find the volume of the volume trap above the cone $z=\sqrt{x^{2}+y^{2}}$ and below the sphere $x^2+y^2+z^2=2$ When I checked the solution I noticed that it was writen as $$V=\int_{0}^{...
4
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2answers
118 views

solution to a general integral $\int_0^\infty \frac{\cos(tx)}{x^2+k^2}e^{-sx}dx$

I would like to find a general solution to the integral: $$I(s,t,k)=\int_0^\infty \frac{\cos(tx)}{x^2+k^2}e^{-sx}dx$$ so far using the substitution $u=\frac xk$ I have managed to reduce this to: $$I(s,...
2
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2answers
69 views

How to evaluate $\int_{0}^{\pi} \sin ^{n}(\eta) d \eta$? [closed]

I have encountered the following integral: $$\int_{0}^{\pi} \sin ^{n}(\eta) d \eta=\underbrace{\left[\left(\sin ^{n-1}(\eta)\right)(-\cos (\eta))\right]_{\eta=0}^{\pi}}_{=0} -\int_{0}^{\pi}\left((n-1) ...
1
vote
1answer
60 views

Problem with the parametrisation of this surface integral

I am facing troubles in understanding (read: "guessing") the correct way to parametrise this integral: $$\int_{\Sigma} \dfrac{1}{\sqrt{1 + x^2 + y^2}}\ \text{d}\sigma$$ Where $\Sigma = \{(x, ...
2
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1answer
44 views

Why are these two upper bounds the same?

I'm trying to upper bound the quantity $\int_t^{\infty} e^{-x^2} dx$ as a function of $t$. I came up with two ways of doing this: We have that $(x-t)^2 = x^2 -2tx + t^2 \ge 0 \implies -x^2 \le -2tx + ...
1
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1answer
26 views

Flux of $F = (3x, y^3, -2z^2)$ through cylinder $x^2 + y^2 = 9$

This is from the UChicago GRE problem sets. Let $C$ be the cylinder bounded by $x^2 + y^2 = 9$ and $z = 0, z = 5$. If $F(x, y, z) = (3x, y^3, -2z^2)$, then calculate the flux of $F$ through $C$, i.e.,...
4
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1answer
40 views

A coin is thrown until two heads and two tails appears. What is the cumulative distribution function of this situation?

A coin is thrown until two heads and two tails appears. Let $Y$ be the number of throws until this happens. What is the cumulative distribution function of $Y$? What I have gotten so far: The last ...
2
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2answers
126 views

Evaluating $\int_{-\infty}^\infty\frac{\cos(2x)}{x^2+4}\:\mathrm{d}x$

As stated in the title, I want to evaluate the integral $$I=\int_{-\infty}^\infty\frac{\cos(2x)}{x^2+4}\:\mathrm{d}x$$ I'm pretty sure it evaluates to $$\frac{\pi}{2e^4}$$ But I'm not sure how to ...
1
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1answer
50 views

How do I evaluate the line integral $\int_c F\cdot dr$

How do I evaluate the line integral $$\int _c F\cdot dr$$ where $F=x^2$i$+2y^2$j and C is the curve given by r$(t)=t^2+t$ j for $t \in [0,1] $. I have started with: $\int f(t^2$i$+t$j$)$$ dr/ dt $ $...
5
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1answer
103 views

Find the area of the region bounded by $\sin(x)\sin(y)=k$ where $0 \leq x \leq \pi$, $0 \leq y \leq \pi$, and $0 \leq k \leq 1$

On the coordinate plane, the equation $\sin(x)\sin(y)=k$ where $0 \leq x \leq \pi$, $0 \leq y \leq \pi$, and $0 \leq k \leq 1$ forms a closed region. Find the area of this region in terms of $k$. I've ...
2
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1answer
59 views

Double integral on a 2D rotated area : $\iint_D (x+y)^3 (x-y)^2 \,\mathrm{d}x\,\mathrm{d}y$

I was assigned an exercise in which I have to calculate a double integral of a given function, in an area which is made from four lines. $D$ is defined by the relations: $x+y=1$ , $x-y=1$ , $x+y=3$ , $...
1
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0answers
53 views

Let $I=\int_{0}^{\pi/2}(\sin 2x)^{1/3}\sin x dx$ $J=\int_{0}^{\pi/2}(\cos 2x)^{1/3}\cos x dx$. Find $I/J$ [duplicate]

I found $I=\int_{0}^{\pi/2}(\sin 2x)^{1/3}\sin x dx$ = $2^{1/2}\int_{0}^{\pi/4}(\cos 2x)^{1/3}\cos x dx$ $J=\int_{0}^{\pi/2}(\cos 2x)^{1/3}\cos xdx$=$2^{1/2}\int_{0}^{\pi/4}(\sin 2x)^{1/3}\sin x dx$ ...
4
votes
5answers
120 views

Estimating $\int_{0}^{1}\sqrt {1 + \frac{1}{3x}} \ dx$.

I'm trying to solve this: Which of the following is the closest to the value of this integral? $$\int_{0}^{1}\sqrt {1 + \frac{1}{3x}} \ dx$$ (A) 1 (B) 1.2 (C) 1.6 (D) 2 (E) The integral doesn't ...
2
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0answers
45 views

Dirac delta function of two variables

I am trying to solve the integral: $$I = \int^{+\infty}_{-\infty} \mathrm{d} x \int^{+\infty}_{0} \mathrm{d} y \frac{f(x,y)}{g(x,y)}$$ where $g(x,y)$ has two zeros: $g(x=a,y)=0$ and $g(x,y=b)=0$, and ...
0
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0answers
30 views

Integration using Derrivative trick

Suppose I have a function $$f(x)=\exp(ax^2+bx+c)$$ and $$\int_{-\infty}^\infty f(x)dx=k(a,b,c)$$ Can we say $$\int_{-\infty}^\infty xf(x)dx={\partial k(a,b,c)\over \partial b}$$ and $$\int_{-\infty}^\...
0
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1answer
36 views

Convergence of the Airy function from Integral by parts

In the book Complex Analysis by Stein, it defines the Airy function as $$ \frac{1}{2\pi} \int_{-\infty}^\infty e^{i(x^3/3+sx)}\ dx \quad \text{for } s \in \mathbb{R} $$ It first says that because of ...
1
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2answers
82 views

Edited: Let $I=\int_{0}^{\pi/2}(\sin 2x)^{1/3}\sin x dx$ $J=\int_{0}^{\pi/2}(\cos 2x)^{1/3}\cos x dx$. Find $I/J$

Edit:Sorry guys I made an error in question, I have edited it now. I found $I=\int_{0}^{\pi/2}(\sin 2x)^{1/3}\sin x dx$ = $2^{1/2}\int_{0}^{\pi/4}(\cos 2x)^{1/3}\cos x dx$ $J=\int_{0}^{\pi/2}(\cos 2x)^...
3
votes
3answers
99 views

How to solve $\int ^{1}_{-1}\frac {x^{2n}}{\sqrt {1-x^{2}}}dx?$

I couldn't solve this. $$\int ^{1}_{-1}\dfrac {x^{2n}}{\sqrt {1-x^{2}}}dx$$ I thought that like the following. $$\int ^{1}_{-1}\dfrac {x^{2n}}{\sqrt {1-x^{2}}}dx=\int ^{1}_{-1}\dfrac {1-\left( 1-x^{2n}...
2
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0answers
53 views

Defining a general structure of “Calculus” [closed]

I've been thinking lately, is there a way to generalize the fundamental concepts of Calculus such as convergence, differentiability and integrability to it's "maximum potential"? That is, ...
8
votes
3answers
168 views

How to integrate $ \int\frac{x-2}{(7x^2-36x+48)\sqrt{x^2-2x-1}}dx$?

How to integrate $$ \int\frac{x-2}{(7x^2-36x+48)\sqrt{x^2-2x-1}}dx\,\,?$$ The given answer is $$ \color{brown}I=-\frac{1}{\sqrt{33}}\cdot \tan^{-1}\bigg(\frac{\sqrt{3x^2-6x-3}}{\sqrt{11}\cdot (x-3)}\...
0
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0answers
20 views

Question about a proof antiderivative of complex function.

Theorem: Let $U\in \Bbb C $ be an open rectangle and $p\in U$. Let $f,g \in C(U,\Bbb R)\cap C^1(U-\{p\},\Bbb R)$. If $$\frac{\partial f}{\partial y}=\frac{\partial g}{\partial x} \;\;\text{on}\;\; U-\{...
-4
votes
0answers
29 views

(a) What is the generalized integral calculation? [closed]

(a) What is the generalized integral calculation? (b) What is the generalized integral calculation for n = 3? (c) What is the generalized integral calculation when n is infinite? $$\int_0^\infty\...
1
vote
3answers
52 views

How to prove the convergence of an integral? [closed]

I need to prove the convergence of the integral $$\int_{1}^{\infty} \frac{x}{(5x^{2}-1)^3}dx $$ I tried all the tests but could not prove its convergence. I was trying the comparison test and was able ...
0
votes
2answers
30 views

Confused about deriving the line element

I wish to use this as my starting point $$ s=\sqrt{x^2+y^2} $$ and then work my way to the line element $ds^2=dx^2+dy^2$, but I cannot understand how to get there. If I derive $s$, I get $$ ds=\frac{...
0
votes
1answer
85 views

Integrating factor in differential equation

We are given a differential equation which involves a integrating factor. the equation is: $$\frac{d I}{d\cos\theta } = \frac{1+ \cos \theta}{1+\sin^2\theta}$$ The way I understand to do integration ...

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