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

Under what assumptions is the following first moment monotone?

I'm working on an economic model and am encountering the following mathematical issue. Let $x\sim \mathcal{N}(\mu,1)$, and define $$V(\mu)=\int_0^{\hat x(\mu)}x ...
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1answer
13 views

Extending the definition of curve length

I know for continuously differentiable curves on closed interval $[a,b]$, the curve length is given by $\Lambda (\gamma)=\int_a^b |\gamma^{'}(t)|dt$. But what about curves such that $\gamma^{'}(t)$ is ...
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4 views

Substitution in complex-valued Fourier integral

In Knapp (Representation theory of semisimple groups, 86'), on page 34 it is shown by means of Euclidean Fourier transform that the principal series representation of $SL(2, \mathbb C)$ is irreducible ...
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0answers
31 views

Find the upper and lower sum of an integral with a floor

I'm having some trouble and looking for some help with a problem i'm trying to solve. Without the floor function it would be easy but the floor has made it a bit trickier: Find the upper and lower ...
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0answers
33 views

Integrating a rational function of exponentials

Let $\gamma ,\mu > 0$ be positive real constants and $\beta \in \mathbb{R}$ be a real constant. How can I evaluate the following indefinite integral? $$ \int \frac{e^{2\gamma t} (e^{-\mu t} - ...
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31 views

Is Randall's model for a sling correct?

In What If 116 Randall Munroe talks about ways to get drivers around a race track The 13th paragraph he imagines the drivers at the end of what seems like a big tether ball: Imagine a "vehicle" ...
1
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1answer
76 views

Integrate 1/ln(ln(x)) asymptotically

I was looking for the asymptotic behaviour of the anti-derivative of $\frac{1}{\ln \ln x}$, in terms of the big-O notation. Wikipedia's list does not have this integral, and Wolfram Alpha says "no ...
2
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1answer
29 views

Fourier Transform of $\delta(t-nt)$

Given the discrete signal $x(n)=\begin{bmatrix} \alpha ^n, n\geq 0 \\0, n<0 \end{bmatrix}$ where $\alpha \in (-1,1)$ and some natural number $N$, we know that the discrete signal $y(n)$ (where $0 ...
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10 views

Let $f:[a,b]\to R$ is continuous and $G(x,t)=t(x-1)$ when $t\leq x$ and $x(t-1)$ when $t\geq x$.

Let $f:[a,b]\to R$ is continuous and $G(x,t)=t(x-1)$ when $t\leq x$ and $x(t-1)$ when $t\geq x$. Let $g(x)=\int_0^1f(t)G(x,t)dt$. Show that $g''(x)$ exists and eqals $f(x)$ for $x \in (0,1)$. I ...
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34 views

Applying contour integration to $\int_{0}^{\pi}dx\frac{cos(x)}{\sqrt{x^2 + x_0^2}}$

Is it possible to apply contour integration to find the value of following integral $$\int_{0}^{\pi}dx\frac{cos(x)}{\sqrt{x^2 + x_0^2}}$$
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63 views

How to compute this triple integral?

I am trying to do this triple integral $$\int_{0}^{\infty }\int_{0}^{\infty }\int_{0}^{\infty }(u+w)e^{-\frac{(u+w)^2}{2}}(v+w)e^{-\frac{(v+w)^2}{2}}(u+v)e^{-\frac{(u+v)^2}{2}}e^{-(\mu +\lambda ...
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29 views

Integral of $\frac{\exp\left(\, -\alpha x\,\right)\, (x-x_0)} {{(x-x_0)^2+\beta^2}}$

Does the following integral have a closed form solution? $$ \int_{0}^{\infty} \frac{\exp\left(\, -\alpha x\,\right)\, (x-x_0)} {{(x-x_0)^2+\beta^2}}{\rm d}x $$ where $\alpha$, $\beta$ and $x_0$ are ...
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1answer
21 views

Why is just $0$ extreme point? v22

We have $f:R\rightarrow R,\:f\left(x\right)=x^3-3x+2$ and we need to find extreme points for $g:R\rightarrow R\:,\:g\left(x\right)=\int _0^{x^2}\:f\left(t\right)e^tdt$. Here is all my steps: ...
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6answers
68 views

$f:[a,b] \to R$ is continuous and $\int_a^b{f(x)g(x)dx}=0$ for every continuous function $g:[a,b]\to R$

$f:[a,b] \to R$ is continuous and $\int_a^b{f(x)g(x)dx}=0$ for every continuous function $g:[a,b]\to R$ with $g(a)=g(b)=0$. Must $f$ vanish identically? Using integration by parts I got the form: ...
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1answer
71 views

Why $\int _0^{x^2}e^{-t^2}dt$ is positive for $|x|>1$ [on hold]

Why $\int _0^{x^2}e^{-t^2}dt$ is positive for $|x|>1$ and negative for $|x|<1$ ? I don't understand .. I can't see.. damn it!
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15 views

Compute the integral $y^2dx -2xydy$ with the curve of an equiliateral triangle with vertices at $(0,0), (2,0)$ and $(1,-2)$

The partial derivates are equal so I thought to use the theroem of line integrals but I also thought using Green's Theorem would be easier or possibly evaluating the each segment of the traingle ...
2
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1answer
24 views

Find the inverse fourier transform of simple function

Suppose that the fourier transform of a signal $x(t)$ is $\hat x(u)=\frac{1}{2u_m}(1+\cos (\frac{\pi u}{u_m}))$ where $u_m \geq |u|$.$t$ here stands for time so $t \geq 0$ We sample the original ...
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0answers
17 views

Computing the value of a line integral of a vector field in the plane

We are given the vector field $ x^2dx+y^2dy $ and are interested in the line integral of it over the closed equilateral triangle with vertices (0,0) (2,0) (1,-2) Because the partial derivatives of ...
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3answers
118 views

Intriguing Indefinite Integral: $\int ( \frac{x^2-3x+\frac{1}{3}}{x^3-x+1})^2 \mathrm{d}x$

Evaluate $$\int \left( \frac{x^2-3x+\frac{1}{3}}{x^3-x+1}\right)^2 \mathrm{d}x$$ I tried using partial fractions but the denominator doesn't factor out nicely. I also substituted ...
2
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1answer
26 views

How can I find monotonicity intervals? v18

We have $F:\mathbb{R}\rightarrow \mathbb{R}$, $F(x)=x\int _0^x (1+\cos(t)) \, dt$ and we neeed to find monotonicity intervals and I don't know how... Here is what I try to do: $$F'(x)=\int _0^x ...
2
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1answer
32 views

Where do the step function integral boundaries come from?

EDIT: I have a confusion about Heavyside step function. Suppose I have integral like $$ \int_{0}^{\infty}dE_1\int_{0}^{\infty}dE_2\int_{0}^{\infty}dE_3 \delta(2- \gamma-E_1-E_2-E_3) $$ my first ...
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2answers
21 views

How we can prove that $a_n=\sum _{k=1}^nf\left(k\right)-\int _0^n f(t)\:dt$ is convergent?

We have $f:\left(-1,\infty \right)\:\rightarrow \:R,\:f\left(x\right)=\frac{x}{x+1}$ and we need to prove that: $a_n=\sum _{k=1}^nf\left(k\right)-\int _0^n\:f\left(x\right)dx$ is convergent.Maybe, in ...
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2answers
57 views

Can somebody please show me the necessary steps to solve this Calculus problem?

I have a homework assignment that asks me to solve the differential equations and it gives me: \begin{align*} xy^2y' & = 2-x\\ y''+4y & = 8x\\ y(1)& =1 \end{align*} Are these three ...
2
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3answers
68 views

Evaluation of $ \int_{0}^{\frac{\pi}{4}}\left(\cos 2x \right)^{\frac{11}{2}}\cdot \cos xdx $

Evaluate $$\displaystyle \int_{0}^{\frac{\pi}{4}}\left(\cos 2x \right)^{\frac{11}{2}}\cdot \cos x \,dx .$$ $\bf{My\; Try::}$ Let $$\displaystyle I = \int \left(\cos 2x \right)^{\frac{11}{2}}\cdot ...
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2answers
43 views

Why Riemann sum is convergent? [on hold]

Why $\frac{1}{n}\sum _{k=1}^nf\left(\frac{k}{n}\right)$ is convergent? I don't understand how we can prove that is bounded and monotone... For instance: $f:R\rightarrow R,\:\:f=\frac{1+x}{1+x^2}$, ...
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2answers
43 views

Two indefinite integral problems. [on hold]

Please help me out in these $$\int \frac{dx}{1-3\sin (x)}.$$ Second $$\int \sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}dx.$$
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0answers
17 views

Is there any closed form for the integration of multiplication of two multivariate normal probability distributions?

I already computed the following integration but its a messy thing. I wonder if there is any easy way to compute it? or it has any closed form? V and p are known where V and p (p<1) are positive. ...
2
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2answers
52 views

Evaluation of $\int\frac{1}{x^2.(x^4+1)^{\frac{3}{4}}}dx$

Evaluation of Integral $\displaystyle \int\frac{1}{x^2\left(x^4+1\right)^{\frac{3}{4}}}dx$ $\bf{My\; Try::}$ Let $\displaystyle x = \frac{1}{t}\;,$ Then $\displaystyle dx = -\frac{1}{t^2}dt\;,$ ...
3
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1answer
22 views

Integrating a sphere by discs vs shells (spherical coordinates)

I am getting very confused about the following. Let's say I want to find the volume of a sphere. I can start with a circle having circumference $2\pi R\cos\theta$. I can multiply by $R d\theta$ and ...
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3answers
40 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 ...
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2answers
49 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? ...
1
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1answer
47 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
25 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
11 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
34 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
53 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
48 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
65 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
16 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
35 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
63 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
49 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
23 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
17 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
44 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 ...
1
vote
1answer
33 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
40 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 ...