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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69 views

how to calculate this line integral $\int_{0}^{2\pi} (16\sin^2 3t +16\cos^2 4t)\sqrt{(144\cos^2 3t +256\sin^2 4t)}dt$

I am working on a line integral to calculate the amount of chocolate to cover a pretzel. the density of the pretzel is given by this formula $\lambda=3(x^2+y^2)$ and the parameter equation of a ...
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48 views

Prove the function is integrable

For a point $x \in [1,2]$, define $f(x) = 0$ if $x$ is irrational and define $f(x)= \frac 1n$ if $x$ is rational and is expressed as $x = \frac mn$ for natural numbers $m$ & $n$ having no common ...
3
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56 views

Fatou's Lemma and Counting Measure

I have a vague problem in a Measure and Integration book here. They ask me to consider $\mu$ the counting measure in $\mathbb{N}$ and interpret Fatou's lemma, monotone and dominated convergence ...
3
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74 views

Klein-Gordon field commutator integral?

Consider a Klein-Gordon field $\phi$, which satisfies $$(\Box+ \omega_0^2)\phi=0$$ on points $x \equiv \{x_0,\vec{x}\},y\equiv \{y_0,\vec{y} \}$ of 4D Minkowski-spacetime. The field commutator is $$ ...
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63 views

Differentation uder the integral sign

Let $F(x)=\int_{\sin x}^{\cos x} e^{x\sqrt{1-y^2}} \, dy $. My task is to calculate $F'(x)$. My idea is to use http://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign and I get: ...
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87 views

Integration involving square root function

First , i multiply numerator and denominator by (2-x-x^2)^(1/2), then I split the integral into 2 parts , using trig substitution , part 2 is easy to be integrated .. but when I tried to ...
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107 views

Is there a way to exploit local redundancy in a function to speed up Monte Carlo integration?

In every Monte Carlo method I've ever seen, $f$ must be recomputed from scratch for each point that is (somehow randomly) selected to contribute to the overall integral. However, most functions have ...
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59 views

Solving system of first-order PDEs with Frobenius theorem

I've been stuck trying to solve this system: $$\ \frac{\partial u}{\partial x} = \frac{-2xy^2}{u} + 3y $$ $$\ \frac{\partial u}{\partial y} = \frac{-2x^2y}{u} + 3x $$ Which must satisfy $ \ u(0,0) = ...
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26 views

Finding the Area of a Torus-like surface

I'm trying to find out the Area of the following surface: Let $C$ be the curve associated to a regular, simple path $\theta:[0,l]\rightarrow \Bbb R^2 $; also assume that ...
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77 views

Prove that primitives of $\frac{x^3}{{\rm e}^x - 1}$ have no closed form in terms of elementary functions

It is known the following indefinite integral $$\int \frac{x^3}{{\rm e}^x - 1} dx$$ cannot be evaluated in closed form in terms of any of the elementary functions of mathematics. A proof of this can ...
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27 views

Correct integral to compute volume of a solid

I want to compute the volume obtained by rotating the bounded region of $y=3-x^2$, $y=2x$, $x \leq 0$ around the $y$-axis. I want to use the cylindrical shell method, so my integral is: ...
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111 views

Integrate $\int\tan (x)\sin (ax)dx$

$$\int\tan (x)\sin (ax)dx$$ If $a$ integer number I used the wolfram.Alpha.com site to give me the following How can I use this result if I need to compute some values by using a program in ...
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51 views

Finding the length of an elliptical spiral

Okay, i had a very strange thought, it was "Is it possible to find the length of an elliptical spiral whose major and minor axes were decreasing?" Like for example lets say that $$ \frac{a}{b} = n ...
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33 views

Analysing an integral with its inverse integral [Image, and good explanation inside]

I have a function $f(x) = \dfrac{x}{\sqrt{1+x^2}}$ This means $f^{-1}(x) = (\pm)\dfrac{x}{1-x^2}$, where the negative solution is ignored for this problem. If I want to find a relation between the ...
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95 views

Show that $\int_0^\infty t e^{-xt}\log(1+x^2) \, dx=(\pi-2\mathrm{Si}(t))\sin t-2\mathrm{Ci}(t)\cos t$, $\forall t>0$.

How to show that $$\int_0^\infty t e^{-xt}\log(1+x^2) \, dx=(\pi-2\mathrm{Si}(t))\sin t-2\mathrm{Ci}(t)\cos t$$ for all $t>0$?
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62 views

Fubini's theorem application proof check

I have proven a problem but I am unsure whether it is correct because the proof seems so simple that I think I might be mistaken. Please be kind to comment on my proof and tell me whats wrong with it. ...
3
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48 views

Integration with respect to signed measure and Radon Nikodym theorem

I aim to show the following question: Let $\mu$ be a $\sigma$-finite measure, and $\lambda$ a finite signed measure on $(X,M)$ satisfying $\lambda\ll\mu$, let $h=\frac{d\lambda}{d\mu}\in ...
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31 views

Is there any relation between summation and indefinite integration?

Indefinite integral is the family of all its primitives or antiderivatives. It represents geometrically a family of curves having parallel tangents at their points of intersection with the lines ...
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36 views

How to take this integral?

How to evaluate $$\int_0^\infty e^{-w^2\cos ^2 x-x} \, dx$$ I have no clue, indefinite integral also would be a help.
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116 views

Evaluate $ \int_0^3 \frac{x^3}{1-x^4}\, dx. $

Evaluate $$ \int_0^3 \frac{x^3}{1-x^4}\, dx. $$ I evaluated the integral and got $\left[\dfrac{-\ln(1-x^4)}{4}\right]_0^3$ which ended up diverging. Any help is appreciated!
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105 views

Using complex analysis, calculate $I_m = \int_{-\infty}^\infty \frac{dx}{1+x+x^2+\cdots+x^{2m}}$ for $I_2$ and $I_3$

Question: Using complex variables, calculate $I_m = \int_{-\infty}^\infty \frac{dx}{1+x+x^2+\cdots+x^{2m}}$ for $I_2$ and $I_3$. Attempt: With some help, I have determined that the integral is ...
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71 views

A hard integral from probability theory

I am trying to resolve this integral, which comes out of considering a compound distribution of normal variables: $$ \int_{-\infty}^{\infty} \frac{1}{\sigma_{\sigma} \sqrt{2 \pi}} ...
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51 views

How to find the Maclaurin series for the integral of $e^{x^2}$?

I am trying to find the Maclaurin series for the integral of $e^{x^2}$? What I done so far is that the Maclaurin series for $e^{x^2}$ is $$e^{x^2}=\sum_{n=0}^{\infty}\frac{x^{2n}}{n!}$$ So would ...
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54 views

Which integrals can be solved using Feynman's Technique?

How to check whether an integral can be easily solved using Feynman's approach. What are the main criteria needed to be taken into account?
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57 views

How to evaluate the following integrals

$$\int\limits_0^{\frac{\pi }{2}} {{x^2}{{\ln }^2}\left( {\sin x} \right)\ln \left( {\cos x} \right)dx} ,\int\limits_0^{\frac{\pi }{2}} {x\ln \left( {\sin x} \right){{\ln }^2}\left( {\cos x} \right)dx} ...
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40 views

Recovering a kernel from a system of equations

Suppose $f\in C([0,\frac{3}{4}]^2)$ and $$\begin{array}{rlr}\text{i.}& \int_0^{\frac{3}{4}-x} f(x,y)dy=-\frac{1}{2}x^2+\frac{9}{32}&\forall x\in [0,\frac{3}{4}]\\ \text{ii.}& ...
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30 views

Multiple Integral Substitution Error

I just started learning about the substitution rule for multiple integrals and I decided to give myself an example problem: Calculate $\iint_R{(x^2 + y^2)dA}$ with $R = \{(x, y) \in \Bbb{R} \ |\ 0 ...
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80 views

Equivalence of Lebesgue integral definitions

I'm currently enrolled in a course in integration and functional analysis following Avner Friedman's Foundations of Modern Analysis. However, I noticed that his definition of the Lebesgue integral is ...
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38 views

How to solve integrals of the form $\int u^{-\alpha} e^{-\beta u} du$?

I have to simplify an integral of the form $\int u^{-\alpha} e^{-\beta u} du$, where $\alpha, \beta \in \mathbb{R}^{++}$. Is it a standard integral, or a family that subsumes gamma integrals? Is there ...
3
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163 views

Integration by parts for multidimensional Lebesgue-Stieltjes Integrals

I am concerned with the following problem: I am wondering if there exists any sort of integration by parts formula for a multidimensional Lebesgue-Stieltjes integral. In my case the integral is given ...
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61 views

An integral with a decaying exponential with rational exponent

I was working on some mathematical derivations while I faced this integral: $$\Large \int_0^\infty x^{\alpha-1}e^{-\beta x} e^{-\lambda \left[\frac{x^2}{2x+\eta}\right]}\ \mathrm{d}x \quad .$$ Does ...
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46 views

Approximating this definite integral

I ran into the following integral in my research that I believe has no closed-form solution: $$ I = \int_{s_0}^{s_1} \frac{(\alpha_x s + \beta_x)^{\lambda_x}}{(\alpha_y s + \beta_y)^{\lambda_y}} ds ...
3
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74 views

How to evaluate the following two integral combined with anti-trigonometric function and trigonometric function?

\begin{align*} &\int_0^{\frac{\pi }{3}} {\arccos \frac{{1 - \cos x}}{{2\cos x}}dx} \\ &\int_0^{\frac{\pi }{2}} {\arccos \sqrt {\frac{{\cos x}}{{1 + 2\cos x}}} dx}. \end{align*} A few days ...
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59 views

Integration indefinite integral of multiple functions

I need help integrating $$\frac{x}{1-\exp(-x^2/a^2)}\exp((x-u)^2/2s^2)$$ wrt $x$, where $a$ and $u$ are constants
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123 views

Is this proof correct? Divergence of $\int_{1}^{\infty} \left| \frac{\sin x}{x} \right| \, \mathrm{d}x $

Problem: Show that $$ \int_{1}^{\infty} \left| \frac{\sin x}{x} \right| \,\mathrm{d}x $$ diverges. I know that there are many questions in which this problem is solved, but I want to know if my ...
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89 views

Dealing with absolute values after trigonometric substitution in $\int \frac{\sqrt{1+x^2}}{x} \text{ d}x$.

I was doing this integral and wondered if the signum function would be a viable method for approaching such an integral. I can't seem to find any other way to help integrate the $|\sec \theta|$ term ...
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127 views

Residue Integral: $\int_0^\infty \frac{x^n - 2x + 1}{x^{2n} - 1} \mathrm{d}x$

Inspired by some of the greats on this site, I've been trying to improve my residue theorem skills. I've come across the integral $$\int_0^\infty \frac{x^n - 2x + 1}{x^{2n} - 1} \mathrm{d}x,$$ where ...
3
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123 views

Help on the Integration of $\int_0^{\infty} e^{-bx}\sin ax^2 \, \mathrm{d}x$.

I have had the misfortune of coming across the following integral, for real $b$ and $a > 0$: $$\int\limits_{0}^{\infty} e^{-bx} \sin\left(ax^{2}\right) \, \mathrm{d}x.\tag{1}$$ Naturally, I ...
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82 views

Integral substitution paradox

Assume $f \in L^+(\mathbb{R})$ and $x>0$. Consider the integral $$ \int_0^\infty \frac{f\left(\frac{x}{y}\right)}{y} \: dy. $$ I am trying to make the substitution $u=x/y.$ I seem to get $$ ...
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142 views

What types of integrals cannot be solved using improper Riemann-Stieltjes Integration?

I came across the wikipedia discussion of the Riemann-Stieltjes integral. The first sentence in the "Generalization" section gave me pause: An important generalization is the Lebesgue–Stieltjes ...
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68 views

Parameter-dependent integral: Is the following statement true?

Is the following statement true? If so, could anyone provide a reference? Suppose $f(x, \alpha)$ is continuous on $(a, b) \times \{\alpha_0\}$. If there exists $g(x)$ which is continuous on $(a, b)$, ...
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41 views

Prove that $I = \int_0^{m(m+1} y_n(x)\,\mathrm{d}x$ converges and $I \in \mathbb{Q}$.

My problem is stated as follows Let $y_0(x) = x, \ \: y_1(x) = \sqrt{x}, \ \: y_{n+1}(x) = \sqrt{y_n(x) +x\,} \ $. Now define $ \displaystyle \hspace{3cm} I_n = \int_0^k ...
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227 views

On an application of the Abel-Plana formula

Referring to a previous question, i am having a hard time trying to do the integral: $$f(s)=-i\int_{0}^{\infty}\frac{\log \left[1+\frac{\left(s\log(1+ix) \right )^{2}}{4\pi ^{2}} \right ]-\log ...
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78 views

Integrating $xe^{a/x^2 - x^2}\text{Erfi}(x/\sqrt{2})$?

I want to solve any of the two integrals for the complex number $a$ \begin{aligned} I_1 & = \int\limits_{0}^{\infty} xe^{a/x^2 - x^2}\text{Erfi}(x/\sqrt{2}) dx\\ I_2 & = ...
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73 views

Finding the integral of $\cos\theta \cdot dt$ in terms of the integral of $\sin\theta \cdot dt$

I have an integral as follows: $$\int_0^T \cos\theta\cdot dt = xT$$ where $\theta$ is a function of $t$ I also have, $$\int_0^T \sin\theta\cdot dt = y$$ I want to solve for $T$. If the ...
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72 views

Properties and representations of the the rescaled complementary error function $\mathrm{erfcx}{z}$

Consider the rescaled complementary error function: $$ \mathrm{erfcx}(z) = {e^{z^2}} \left( {1-\mathrm{erf}(z)} \right) $$ $z \in \Bbb{C}$ which also has the following integral representation: $$ ...
3
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59 views

Does this integral variable change makes sense to you?

I was Reading a book about calculus when I've found this part about variable substitution in integrals: Consider $f$ defined in na interval $I$. Suppose that $x =\phi(u)$ is inversible, and its ...
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84 views

Strong Integral

Is there a strong(!) notion of integral that can face all of those issues: Singularities Decay Modes Oscillations Measure Spaces Locally Convex Spaces For example combining decay modes with ...
3
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36 views

Evaluating a triple integral explained step by step

Evaluate: $$ \iiint_{D}\sqrt{(1-9z^2)(1-4y^2-9z^2)}\,dx\,dy\,dz$$ where $D$ is the domain: $$D: x^2 +4y^2+9z^2\le1$$ Can someone tell me if my steps are correct? $$\int_{\frac{-1}{3}}^{\frac{1}{3}} ...
3
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39 views

How could i find the pdf of exponential distribution from its characteristics function?

I know that the characteristics function of the exponential distribution is as following: $$ \phi_x(t) =\frac{\lambda}{(\lambda -it)}$$ Also, I know that the pdf of the exponential distribution is: ...