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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90 views

Application of integrating $\cos^4 x$?

A student asked a colleague the other day for a practical application that involved needing to integrate the fourth power of cosine, but no one here could think of one off-hand other than some volume ...
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1answer
33 views

Differentiating a function and using the result to calculate the indefinite integral of another.

We should differentiate the function $f(x) = \sqrt{cosx}$ and use the result to calculate the indefinite integral $\int \frac{sinx}{\sqrt{cosx}}dx$. So I started by differentiating $f(x) = ...
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1answer
283 views

Product of two exponentially distributed random variables

I am trying to find the close form expression of probability distribution of $Z$ such as $Z=X_1X_2$ where $X_1$ and $X_2$ are two independent exponentially distributed variables with PDF ...
2
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0answers
48 views

An upper Bound for $(f(a))^2$, $a\in[0,1]$ in terms of $\int_0^1(f(x))^2dx$

Is there any way to find an upper Bound for $(f(a))^2$, $a\in[0,1]$ in terms of $\int_0^1(f(x))^2dx$. There is a commonly used upper bound in terms of $\int_0^1(f_x(x))^2dx$, but I do want to make ...
1
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0answers
38 views

Integration by Parts in matrices

Given Data in the question We have a given equation based on matrices as follows $\frac{\mathrm{d} R(s)_{3\times3}}{\mathrm{d} s}=R(s)_{3\times3}K(s)_{3\times3} \tag 1$ $\frac{\mathrm{d} ...
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3answers
38 views

$x e ^{-\frac{x^2}{2}}$ at $\infty$ to $-\infty$

I want to know how to explain $\left. \left(x e ^{-\frac{x^2}{2}} \right) \right|_{- \infty} ^{\infty}$ is zero? Is it because the speed of exponentiation is greater than that of linear? How to ...
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0answers
29 views

multivariate quadrature

Assume that $f:\mathbb{R}^n\ \to \mathbb{R} $. We want to approximate the integral, $\int_{I_d} f \, d\mu$. Let $U^{m_i}$ be a quadrature rule in $x_i$ in direction of $x = (x_1 , \dots , x_n)$, with ...
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2answers
25 views

What would the next step be in this integral problem?

I have to integrate the following $\int \frac {3x^2+x+5}{3x^2+x+4}dx$ what I can see is that i can substitute $3x^2+x+4$ for $u$ thus making my integral $\int \frac {u+1}{u}du$ this will give me $\int ...
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0answers
44 views

how we integrate function s respect to y?

I would like to integral this function $$ \int\exp\big(x^2-y^2\big) \Big( \!2y\cos(2xy) +2x\sin(2xy)\Big)\mathrm{d}y $$ Thank you! $\exp\big(x^2-y^2\big)$ is a common factor for sin and cos
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1answer
66 views

Evaluating $\int x^n e^{x}dx$

I consider, for $n=0,1,2,...$, $$ u_n(x)=\int x^n e^{x}dx.$$ I've performed an integration by parts giving $$ u_n(x)=nx^{n-1} e^{x}-nu_{n-1}(x).$$ I'm looking for a closed form. Thank you.
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0answers
37 views

Derive Runge-Kutta matrix with known weights and nodes

Can I derive a Runge-Kutta method by choosing freely the weights and the nodes? what are my constraints? So, if this is the general form of the explicit RK method: $$ y_{n+1} = y_n + \sum_{i=1}^s b_i ...
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0answers
26 views

Find function that gives Fourier transform value of 1

I am trying to find a function $f(a)$ so that the following expression $$ f(a) e^{-\frac{1}{2}\frac{x^2 + y^2}{a^2}} $$ has a Fourier transform equal to 1 as $a \rightarrow 0$. The reason I am doing ...
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0answers
45 views

The partial derivatives of the function $s=\int _u^v\frac{(1-e^t)}{t}dt$

If $s=\int _u^v\frac{\left(1-e^t\right)}{t}dt$, I want to find $\frac{∂s}{∂v}$ and $\frac{∂s}{∂u}$ and their limits as $u$ and $v$ tend to zero. first i find for $\frac{∂s}{∂v}=\frac{∂}{∂v}\int ...
3
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4answers
57 views

What do limits of functions of the form $te^t$ have to do with l'Hopital's rule?

I have an improper function that I have to integrate from some number to infinity. Once integration is done, the function is of the form $te^t$. What I'm wondering is what does this have to do with ...
1
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1answer
73 views

Green-Riemann Theorem

I calculated the circulation of the vector field : $$\vec{v} = -y\omega \, \vec{i} + x\omega \, \vec{j}$$ over the ellipse : $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ I found $2 \pi \omega a b$. ...
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1answer
28 views

On the equivalency of two indefinite integrals using u substitution.

I am reading the Separation of variables page on wikipedia, at a certain point it states that the following equation Is equal to (1) because of the substitution rule of integrals. The ...
1
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1answer
51 views

Help explaining divergence theorem example

I am looking at an application of the divergence theorem, and I don't understand what's going on. Could anyone explain how to go from the first expression to the second expression (which can then be ...
5
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1answer
39 views

Integral versus hypergeometric series: how to solve this?

How can I resolve the following indefinite integral using hypergeometric series? $$ \int (x^3 + 1)^\frac{1}{3} \,dx $$ Wolfram Alpha indicates that the series of Appell are used, but how to get to ...
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0answers
65 views

Fourier Cosine Series question

If I have even piecewise periodic function ($T=6$) $$x(t)=\begin{cases} 0 &-3\leq t \leq-2  \\ 2+t &-2\leq t \leq-1 \\ 1 &-1\leq t \leq 1 \\ -t+2 &1\leq t \leq 2 \\ 0 &2 \leq ...
1
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2answers
49 views

Logarithmic part of the Risch Alorithm

I'm reading some paper about the Risch algorithm and wanted to try a little example: I want to find an elementary solution for: $$\int\frac{1}{e^x + 1}$$ The following lemma tells me how to do ...
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1answer
30 views

Cosine fourier series integration

If I have even piecewise periodic function ($T=6$) $$x(t)=\begin{cases} 0 &-3\leq t \leq-2  \\ 2+t &-2\leq t \leq-1 \\ 1 &-1\leq t \leq 1 \\ -t+2 &1\leq t \leq 2 \\ 0 &1 \leq ...
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0answers
75 views

Gauss /divergence theorem: generalization

The divergence theorem / Gauss integral theorem states that $\int dV\; \nabla \cdot \vec F = \int dS\; \hat n \cdot \vec F$ for a vector function $\vec F$, with $dV$ the volume element, $dS$ the ...
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2answers
59 views

What is the relation between two integrals?

Let us suppose that we have two integrals, $I_1$ and $I_2$ with the same non negative integrand. The integration contour of $I_1$ is a subset of the the integration contour for $I_2$. What can we say ...
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1answer
50 views

Integrals such as x+y wrt x

This question is in lieu of "Integrating $dψ=(x+y)dx+x_0dy$". Though I got good answers, none of them could explain the real question. Consider the integral, $\int(x+y)dx$ In a book I recently read, ...
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0answers
29 views

Integration contour as points of set.

If B is the subset of A, I wounder do we have two integration contour or one for this? What will happen if we take AUB i.e. A union B, than do we have one integration contour? and what if we take A ...
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1answer
223 views

I wanna know another method of Integration int 1/(a+bsinx) dx

$$\int_{0}^{2\pi} \frac{1}{a+b\sin(x)} dx = \frac{2\pi}{\sqrt{a^2-b^2}} ~ \text{if} ~ a^2 > b^2 $$ I know the trick substituting $y=\tan(x/2)$ But I'd like to know another method. For example ...
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2answers
92 views

$\int_{23\pi}^{71\pi/2}\ln \left ( \frac{\left ( 1+\sin x \right )^{1+\cos x}}{1+\cos x} \right )\,dx$

I ran into this integral and I'm trying to solve it. $$I=\int_{23\pi}^{71\pi/2}\ln \left ( \frac{\left ( 1+\sin x \right )^{1+\cos x}}{1+\cos x} \right )\,dx$$ Well, this has something to do with ...
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6answers
169 views

Prove that $ \int_0^{\pi} \frac{(\cos x)^2}{1 + \cos x \sin x} \,\mathrm{d}x =\int_0^{\pi} \frac{(\sin x)^2}{1 + \cos x \sin x} \,\mathrm{d}x $

In a related question the following integral was evaluated $$ \int_0^{\pi} \frac{(\cos x)^2}{1 + \cos x \sin x} \,\mathrm{d}x =\int_0^{\pi} \frac{\mathrm{d}x/2}{1 + \cos x \sin x} ...
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1answer
46 views

Compute integral containing a matrix

Let $\mathbf{H}= \begin{pmatrix} h_{11} & h_{12} \\ h_{21} & h_{22} \end{pmatrix}$ and $P(\mathbf{H}$) the joint probability distribution of $\mathbf{H}$ given by: $e^{-(a+ ...
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0answers
23 views

Partial Derivative of a function containing two Integrals

Looking for some pointers as to how to perform a Partial Derivative of a function that contains two Integrals. I think it's the Leibniz's Rule but not sure... If we are given the following function: ...
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1answer
23 views

Definite Integrations involing volumes

suppose the area of the region is bound by the curve y= (e^x + e^-x)/2 , the lines x=-1 and x=1 and the x-axis. Find the volume of the solid of revolution obtained when the region is rotated about the ...
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1answer
48 views

How is Fubini Theorem used here?

Let $\mu$ be a $\sigma$-finite translation invariant measure defined on the Borel subsets of $\mathbb R^d$ and $\lambda$ be the usual Lebesge measure. My question is how the Fubini theorem is used in ...
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2answers
113 views

Finding volumes with double integrals

I got some troubles with this problem: A swimming pool is circular with $20\,ft$ diameter. The depth is constant along east-west lines and increases linearly from $4\,ft$ at the south end to $9\,ft$ ...
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0answers
13 views

Limit of a quotient of volumes

Let $f: \Omega \subseteq \mathbb{R}^n \to \mathbb{R}$ be a $C^1$ diffeomorphism defined on a compact subset $\Omega$. I want to see why for any $a \in \Omega$: $\lim_{r \to 0} ...
2
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1answer
42 views

Differentiation under integral sign without DCT

Suppose $f: \Omega \times I \subseteq \mathbb{R}^n \to \mathbb{R}$ is differentiable, where $\Omega$ is measurable and $I$ is an open interval. How do you show that if $\frac{\partial f}{\partial t}$ ...
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2answers
110 views

double integral over an arbitrary triangle

Assume we have an arbitrary triangle ABC in x-y plane and we want to integrate a function $f(x,y)$ over surface of this triangle as shown in fig. 1: We can define another coordination system [x' ...
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3answers
81 views

Integrate $\frac{x}{1+x^2}$

Integrating $\frac{x}{1+x^2}$ becomes $\ln \sqrt{x^2 + 1}$ Why is this? Is there a formula or a fact that makes this so. Integrating this a lot different than integrating something easy like $X^2$.
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2answers
103 views

Which integral is larger?

The question: Given $f$ to be a positive, measurable function on $[0,1]$, which is larger, $\displaystyle\int_0^1 f(x)\log f(x)\,dx$ or $\displaystyle\left(\int_0^1f(s)\,ds\right)\left(\int_0^1\log ...
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0answers
34 views

Normalisation of Monte Carlo overlap

Background: In quantum mechanics you are sometimes required to compute the overlap (inner product) of two wave functions (square integrable complex functions) $\Psi(x)$ and $\Phi_i(x)$ as $$ ...
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1answer
20 views

Integrability condition for bounded function defined in a block

I need a hint for this question: I need to prove that if $f: A \subseteq \mathbb{R}^n \to \mathbb{R}$ is bounded, where $A$ is an $n$-dimensional box, then $f$ is (Riemann) integrable iff for each ...
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2answers
118 views

Complex Measures: Decomposition

Given a complex measure: $\mu:\Sigma\to\mathbb{C}$. Consider its decomposition into positive measures: $$\mu=\Re_+\mu-\Re_-\mu+i\Im_+\mu-i\Im_-\mu=:\sum_{\alpha=0\ldots3}i^\alpha\mu_\alpha$$ ...
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1answer
43 views

Trapezoidal rule over interpolation of higher dimensional vectors

According to a wikipedia and mathworld, the trapezoidal rule is: $$ \int_a^b f(x)\,dx \approx h\left[\frac{f(a) + f(b)}{2} \right], $$ where $h = (b-a)$. If you apply this rule to a function ...
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1answer
320 views

how to calculuate $\int_0^ \pi \sqrt{1+x^2 \sin^2x}dx$

I was finding arc length of $y=\sin x - x \cos x$ $(0 \leq x\leq \pi)$ and I found I've to solve $$\int_0^\pi \sqrt{1 + x^2\sin^2{x}}\, dx $$ but I have no idea about this. I tried using $\sin^2x ...
3
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4answers
109 views

Integration without substitution of $\frac{x^2+3}{x^6\left(x^2+1\right)}$

This is a repost of a question i had written incorrectly earlier. How do I integrate this without substitutions ? $$ \frac{x^2+3}{x^6\left(x^2+1\right)} $$ I got: $$ ...
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0answers
49 views

Integral Hölder bound

I was wondering if it is possible to find the following bound or if not, find a counterexample of it. Let $f\in C_0^1$ (compactly supported continously differentiable, in particular $\alpha$-Hölder ...
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0answers
80 views

Liouville's Extension of Dirichlet Theorem

Can we use Liouville's Extension of Dirichlet Theorem to find triple integral $\int\int\int(u^2+v^2+w^2)\space du\space dv\space dw\space where\space u=0, v=0, w=0\space \&\space u+v+w\leq1$? Or ...
2
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1answer
101 views

Resources to investigate rational numbers

I have been told that resources like Mathematica's Number Recognition (which I've never tried myself) and the Inverse Symbolic Calculator (ISC) can be used to find possible closed forms for real ...
4
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2answers
102 views

Integration without substitution

How to i integrate this with out substitutions or Partial fraction decomposition ? ($3x^2$+$2$)/[$x^6$($x^2$+1)] I've got to : 2/x^6(x^2+1),but after this i haven't been able to eliminate the 2.
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4answers
53 views

Definite Integrations problems [closed]

If $f(x)= x^2 e^{x^2}$ then show that $f'(x)= 2xe^{x^2} + 2x^3 e^{x^2}$ and use this result to evaluate $$\int x^3 e^{x^2} \, dx$$ How can I use the result to evaluate the integral?
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2answers
24 views

Behaviour of the error function as $z \rightarrow -\infty$?

I'm trying to find the behaviour of the error function, $erf(z)$ as $z \rightarrow -\infty$ $$erf(z) = \frac{2}{\sqrt{\pi}}\int_0^{z} e^{-s^2}\mathrm{ds}$$ I know that we can find the limit of ...