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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9 views

Evaluate the integral in terms of areas.

I understand that the first one is 4 from basically adding the squares inside the signed area, but I'm unsure on how to proceed in getting the other integrals. Any help would be appreciated, thank ...
2
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2answers
44 views

If $f, g \in L^p$, is it true that $\int | f g | = \int | f | \int | g |$?

Let $f,g \in L^p(0, 1), \;\; 1 < p < \infty$. In this case, is it true that $$\underset{(0, 1)}{\int} | f(x) g(x) | dx = \underset{(0, 1)}{\int} | f(x) | dx \underset{(0, 1)}{\int} | g(x) | dx? ...
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1answer
41 views

Can the following nonlinear first order ODE be solved?

I have tried solving this equation from several manners but no luck. Can it be solved? $$\frac{d f}{d t} = A f^2 +g(t)$$ The solution for the homogeneous is (I think; somebody should confirm) ...
1
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1answer
14 views

Heaviside function & Integral Limits

When considering integration, how does one use the Heaviside function in order to alter the limits of integration. For example If i have $$ \int_a^b f(x) dx $$ But want to change this integral to be ...
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0answers
9 views

Integration of $dR/(d{\alpha}R_0)=-\tan (\pi/N)$

i was doing a physics problem (circular motion's problem) where i had to deal with an object in motion approaching to the center over a regular N-agon and i end up having to integrate this function: ...
1
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1answer
26 views

Double integral of $\arctan(x + y)$?

I would like to find $\int_a^b\int_a^b\arctan(x+y)dydx$ I can "simplify" the integration down to $\int_a^b ((x+b)\arctan(x+b)-\frac{1}{2}\ln(1+(x+b)^2) - ...
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2answers
48 views

Fundamental Theorem of Calculus, application

I want to derive the function $$F(x)=\int_a^{x^2}\sin^3t\,dt$$ with the fundamental theorem of calculus, but I dont know how to handle the $x^2$. Maybe with subsitution I think Fundamental theorem ...
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3answers
45 views

I having trouble using partial fractions.

For the integral below I have to use partial fractions, however I am at a lost on how to do so. $$\int\frac{dt}{t^2-t-20}$$ The farthest I have gotten to is factoring the denominator to $(t+5)(t-4)$. ...
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2answers
58 views

Finding $\int \frac{1+\sin x \cos x}{1-5\sin^2 x}dx$

Find $\int \frac{1+\sin x \cos x}{1-5\sin^2 x}dx$ I used a bit of trig identities to get: $\int \frac {2+\sin (2x)}{-4+\cos(2x)}dx$ and using the substitution: $t= \tan (2x)$ I got to a long ...
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1answer
15 views

About Weierstrass / Tangent half-angle substitution

From: http://en.wikipedia.org/wiki/Tangent_half-angle_substitution How did $\frac 1 {2\cos ^2 \frac x 2}$ become: $\frac {1+t^2} 2$? From the substitution of $\cos x$, it should be similar to: ...
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0answers
11 views

Transforming PDF

I mostly understood the transformation of a pdf for $x$ to a pdf for $y$, which is, given transformation $y(x)$ (and many regularity assumptions) $$ P(x \leq Y \leq b) = \int_a^b f(x) dx \\ P(y(a) ...
2
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0answers
24 views

Asymptotic expression of $\int_{- D}^{D} \frac{\text{tanh}(\xi)}{\xi -\omega}\mathrm{d}\xi$

How to derive the following asymptotic expression ($|\omega| \ll D $)? $${\cal{P}}\int_{- D}^{D} \frac{\text{tanh}(\xi)}{\xi -\omega}\mathrm{d}\xi \approx ...
2
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2answers
59 views

What is the largest function whose integral still converges?

Let C be the set of all functions $f(x)$ whose integral converges, i.e. for some constant $x_0$: $$\int_{x_0}^\infty f(x) dx < \infty$$ While playing with integrals in Wolfram Alpha, I noticed ...
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3answers
43 views

determine $\int x\sqrt{1-x^2}\,dx$

I have to determine $\int x\sqrt{1-x^2}\,dx$ and I have a little question about the substitution. I tried to subsitute $t=1-x^2$. It is $dt=-2xdx$ and therefore $dx=\frac{-dt}{2x}$. But it is the ...
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0answers
14 views

On the centroid of a triangle

There's three different ways to see a triangle in the Euclidean plane: as three non-collinear points, say $A$, $B$, $C$; as the line segments connecting the three points, that we can parametrize as a ...
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0answers
15 views

A question about the condition of quadrature formula

I am reading through my numerical mathematics script and I am currently in the chapter 4 (see listing) computer arithmetic direct solution of linear systems of equations polynomial interpolation ...
0
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2answers
50 views

function which is Riemann integrable

Consider $f:[-1,1]\to\mathbb{R}$, $x\mapsto \begin{cases} 1, & \text{if } x=0 \\ 0 & \text{else } \end{cases}$ I want to know why f is Riemann integrable and I tried something. First of ...
2
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1answer
48 views

Integrating $\int \frac{\sin^3x}{(\cos x)^\frac 4 3} dx$

Find: $\int \frac{\sin^3x}{(\cos x)^\frac 4 3} dx$ My attempt: Set $u=(\cos x)^\frac 4 3 $ so $du= \frac 4 3 (\cos x)^\frac 1 3 \sin x dx \Rightarrow dx= \frac 3 {4 (\cos x)^\frac 1 3 \sin ...
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1answer
42 views

How to show $\int_{x_0-\delta}^{x_0+\delta} g(x) > 0$ if $g(x_0)>0$? [on hold]

(a) Let $f$ and $g$ be Riemann integrable functions on $[a,b]$. Prove that if $f(x)\le g(x)$ for all $x\in[a,b]$, then $$\int_a^b f(x) dx \le \int_a^b g(x) dx.$$ (b) Prove that if $g$ is ...
0
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0answers
17 views

How to generate integer random numbers that equal to another random number?

I am running a simulation in Excel, and need to generate a group of integer random numbers summing up to another random integer, how can I possibly do it? For instance I have an integer random number ...
0
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0answers
14 views

Volume of a conic set

Consider $K$ a convex body in $\mathbb{R}^{n-1}$ and $x$ a point that doesn't lie in the affine clousure of $K$. If we define the conic set $L=conv(K\cup\{x\})$ and we want to calculate it's volume, ...
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0answers
35 views

Common traits of functions which are non-trivial to integrate?

My question is very simple: do there exist certain qualities of functions such that functions which possess these qualities are guaranteed not to have anti-derivatives which are expressable in terms ...
1
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1answer
31 views

Differentiation under the integral

Now I have this expression. $\psi(\theta)=\text{log}\int_{-\infty}^{\infty}\exp{\{\Delta\theta-f(\nu)\Delta^2\}}h(\Delta)d\Delta$. The expression of $h(.)$ is not given. So $h(\Delta)$ is some ...
2
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1answer
31 views

An Application of Stokes's Theorem

Let $D^2=\{(x,y)\in \mathbf R^2: x^2+y^2\leq 1\}$ be the unit disc in $\mathbf R^2$, and $D^3=\{(x,y,z)\in \mathbf R^3: x^2+y^2+y^2\leq 1\}$ be the unit disc in $\mathbf R^3$. Let $i_{\pm}:D^2\to ...
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0answers
17 views

Finding point on ellipse given an arc length

Given a parametric representation of an ellipse: $$ x = a\cos t \\ y = b\sin t $$ Say I have a known point $P_0$ at $t = t_0$. Given also a known arc length $d$ on the ellipse: $$ d = ...
5
votes
2answers
59 views

Evaluate $\text{k}$ from the given equation

If $$ \int_{0}^{\infty} \left(\dfrac{\ln x}{1-x}\right)^{2} \mathrm{d}x + \text{k} \times \int_{0}^{1} \dfrac{\ln (1-x)}{x} \mathrm{d}x =0$$ then find the value of $\text{k}$ My ...
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1answer
34 views

Difficult integral $\frac{du}{u}=\left(\frac{x+y}{x}\right)dx$ in PDE

The linear problem is given as $$x\frac{\text{$\delta $u}\backslash }{\text{$\delta $x}}\text{+y}\frac{\text{$\delta $u}\backslash }{\text{$\delta $y}}\text{=(x+y)u}$$ with $u = 1$ on $x=1$ with ...
0
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1answer
22 views

Express $\sinh(x+y)$ and $\cosh(x+y)$ in terms of $\cosh(x), \sinh(x), \cosh(y), \sinh(y)$

I don't know if my answer is ok or not for this question below: Use the definitions $$\cosh(x)=\frac12\left(e^x+e^{-x}\right), \hspace{0.2in}\sinh(x)=\frac12\left(e^x-e^{-x}\right)$$ to express ...
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0answers
50 views

On definition of Riemann integral

Let $I = [a, b]$ be the finite closed interval of $\mathbb R$. A partion $P$ of $I$ is a finite sequence $a = a_0 \lt a_1 \lt ... \lt a_n = b$. We write $P\le Q$ if $P \subset Q$ where $P, Q$ are ...
4
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2answers
314 views

Is the gamma function expressible as a proper integral?

Is the gamma function expressible as a proper integral of elementary functions? You're also allowed to compose it with however many elementary functions. But strictly no limits. [edit] So far the ...
1
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1answer
47 views

Trigonometric integral evaluates to factorial

I would like to prove the integral identity $$\int_{0}^{2\pi} e^{\cos(x)} \cos(nx - \sin(x)) \, dx = \frac{2\pi}{n!}$$ One approach is to interpret this as the real part of a complex exponential ...
4
votes
2answers
64 views

What are some tips/techniques that might help me solve this (brutal) differential equation?

I've been working on a certain physics problem involving differential equation for two years. I've made some progress on it recently, but I've come across another roadblock, namely an integral that I ...
0
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1answer
52 views

How can I integrate $\frac{1}{x^2-x-1}$?

I need to find $\int\frac{1}{x^2-x-1}dx$ and I don't know what to do. I've thought about substitution or partial fractions but neither has worked.
0
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0answers
7 views

Moment of inertia of lower dimensional bodies and limits

I have noticed that the moment of inertia of an $n$-dimensional body having uniform density $\rho$ is at least sometimes identical to the limit of the moment of inertia of a $(n+1)$-dimensional body ...
0
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1answer
30 views

First order differential equation (with a logistic function)

I came across this first order differential equation $$ f'(x) = \left( \frac{1}{x} + \frac{g'(x)}{g(x)} \right) f(x) - c \frac{g'(x)}{g(x)} \textrm{,}$$ where $g(x)$ is this logistic type function $$ ...
0
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0answers
18 views

Integration - probability

Calculate expected value and variance of the uniform distribution on [8; 42]. I have tried inserting the numbers into the appropriate integrals but I could not come to a solution.
0
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1answer
26 views

even function definite integral

I saw many proofs for the fact $$\int_{-a}^{a}f(x) \, \mathrm dx =2\int_{0}^{a}f(x) \, \mathrm dx$$ for even real function $f$, for example see this. All proofs, which I saw, use the method of ...
1
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2answers
48 views

Evaluating $ \int \frac{1}{5 + 3 \sin(x)} ~ \mathrm{d}{x} $.

What is the integral of: $\int \frac{1}{5+3\sin x}dx$ My attempt: Using: $\tan \frac x 2=t$, $\sin x = \frac {2t}{1+t^2}$, $dx=\frac {2dt}{1+t^2}$ we have: $\int \frac{1}{5+3\sin x}dx= 2\int ...
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0answers
59 views

Integration over a manifold with boundary (Check).

Assume that $ f: \Bbb{R}^{3} \to \Bbb{R} $ is a smooth function such that $ M \stackrel{\text{df}}{=} \left\{ \mathbf{x} \in \Bbb{R}^{3} ~ \middle| ~ f(\mathbf{x}) \ge 0 \right\} $ is a non-empty ...
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0answers
22 views

Measureable functions and its properties

I have two question about measureable functions and its properties and I want some help to solve them $1)$ if $f$ and $g$ are positive measureable functions then $f-g$ is measureable function ? ...
1
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1answer
57 views

Does the improper integral $\int_{0}^{\infty}\sin(x^2)\;\mathrm dx$ converge? [duplicate]

Does the improper integral $\int_{0}^{\infty}\sin(x^2)\;\mathrm dx$ converge? So if it converges then $\lim_{b \to\infty}\int_{0}^{b}\sin(x^2)\;\mathrm dx$ exists and our integral converges to this ...
0
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1answer
26 views

Integration: Finding area, volume and arc length

I am new to integration, so please do not mark this question as "not enough research done" Here is the question (please open image in new tab to see it clearly) - I am getting stuck with the ...
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1answer
34 views

Function is identically zero almost everywhere

Prove that if $\int_E f d\mu = 0$ for some $f \ge 0$, then $f = 0$ almost everywhere. This is Execrise 1 in Chapter 11 of baby Rudin. My attempt: $\int_E f d\mu = 0 \implies$ sup { ${\int_E s ...
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0answers
37 views

Integration help needed with two problems

Here are 2 problems that are confusing me to a great extent - I have no issues with integration, but I'm unable to figure out the limits and the equation to be integrated in these problems. The ...
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1answer
17 views

Quadratic function with positive integral coefficients problem

Here is the problem statement: Let $f(x)$ is a quadaratic expression with positive integral coefficients such that for every $\alpha, \beta\; \epsilon\; \Re$, $\beta>\alpha$, $\int_\alpha^\beta ...
3
votes
1answer
52 views

How can you find the integral of $\frac{cos(2t)}{2t^2}$ between 1 and infinity?

How can you find the integral of $\frac{\cos(2t)}{2t^2}$ between 1 and infinity? $$ I = \int\limits_1^\infty \frac{\cos(2t)}{2t^2} dt $$ My problem is that I just simply do not know how to handle ...
1
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1answer
15 views

Find the length of the curve r(t)= <t^2,2t,lnt> from t=1 to t=e

Find the length of the curve r(t)= $<t^2,2t,lnt> $ from t=1 to t=e i know that Length= $\int$ length of r'(t) dt Therefore, L= $\int _1^e\sqrt{4t^2+4+\frac{1}{t^2}}dt\$$ but i'm having trouble ...
1
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3answers
14 views

Converting unit square domain in (x,y) to polar coordinates

I have the following double integral $\int_{0}^{1}\int_{0}^{1}\frac{x}{\sqrt{x^2+y^2}}dxdy$ The integrand is fairly simple: $\frac{x}{\sqrt{x^2+y^2}}dxdy=\frac{rcos(\theta ...
0
votes
1answer
38 views

Integral - complex exp. term

Does anyone know a suitable method to integrate and/or know the answer to: $\int\limits_{-\pi}^{\pi}$ $\log\Big[\tfrac{2 - a\exp({-it})}{1 - a\exp({-it})}\Big] $ ${\mathrm{d}t}$, for constant $|a|$ ...
1
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3answers
33 views

I have a trouble with an integration from a book of kinetic gas theory.

How integrating this expression: $$ \text{dn}=-n(x) N \pi \left(r_1+r_2\right){}^2\text{dx} $$ in order to evolves into this: $$ n(x)= N_0 e^{ -N \pi \left(r_1+r_2\right){}^2x} $$ Or how ...