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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55 views

Directional derivate of gradient at point with given unit vector

Not entirely sure what I'm doing wrong here, here is the question, it is in Danish though, so I'll post a translation below: $$\mathbf f(x,y) = x^4 + 3x^3y^3 + 6y^2$$ Udregn den retningsafledede ...
2
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5answers
172 views

Evaluating the integral $\int_{-\infty}^\infty \frac{\sin^2(x)}{x^2}e^{i t x} dx$

I want to evaluate the integral $$\int_{-\infty}^\infty \frac{\sin^2(x)}{x^2}e^{i t x} dx$$ for all $t \in \mathbb{R}$. I would preferably do it using the tools of complex analysis, but since I ...
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3answers
14 views

Unspecific boundaries for finding area by double integration

I've been given the boundaries $0≤x≤x^2+y^2≤1$. I have no set equation so it would simply be 1 integrated. Normally I have no problems when the boundaries are clearly divided, yet here I can't seem to ...
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0answers
49 views

Analytical Abel transform of these basis functions

I am trying to perform Abel transforms on basis functions $f_k(r)P_l(cos(\theta))$ for Abel inversion. The typical radial basis functions used are Gaussians $e^{-(\frac{r-r_k}{\sqrt{2}\sigma})^2}$. I ...
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1answer
31 views

is $u\in L_{loc}^1(\mathbb{R}^2)$ and 2. $u\in W_{loc}^{1,1}(\mathbb{R}^2)$?

Let $U\in C^1((0,\infty))$, $u:\mathbb{R}^2\to \mathbb{R}$ defined by $$u(x,y)=U(\sqrt{x^2+y^2}).$$ Under which conditions on $U$ is 1.$u\in L_{loc}^1(\mathbb{R}^2)$ and 2. $u\in W_{loc}^{1,1}(\...
1
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0answers
13 views

Marginalization of joint density with a linear conditional relationship

Let $Y$ and $X$ be two vector-valued random variables and let $Z=f(X)$ implicitly define a third random variable from $X$. $X$ and $Z$ exist in the space $\Omega$. For simplicity, we assume $f(x)=Ax$ ...
2
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1answer
73 views

Closed form of $\int_{x = 0}^{C} \exp\left(-\frac{x}{A}-\frac{B}{x}\right)\,dx$

Is there a closed-form expression for the following definite integral? \begin{equation} \int_{x = 0}^{C} \exp\left(-\frac{x}{A}-\frac{B}{x}\right)\,dx, \end{equation} where $A$, $B$, and $C$ are ...
1
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4answers
46 views

integrate $\int \frac{x-3}{\sqrt{1-x^2}}$

$$\int \frac{x-3}{\sqrt{1-x^2}} \mathrm dx$$ I know that $\int \frac{1}{1-x^2}\mathrm dx=\arcsin(\frac{x}{1})$ but how can I continue from here?
2
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2answers
166 views

Integral of delta dirac function

I try to calculate the following integral: \begin{equation} \int^{+\infty}_{-\infty} \frac{x^4 \exp{(ixa)}}{1+x^2b^2} \mathrm{d}x \end{equation} where $a,b$ are real positive numbers. This integral ...
3
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2answers
95 views

Is there a simple proof for $\int_1^{\infty}\frac{2x^2\log^2 x}{(x^2-1)^2}dx=\frac{1}{4}(7\zeta(3)+\pi^2)$?

This morning I've computed easy computations with simple integral representations for Apéry constant and I find a (conjecture) formula using an online integrator (Wolfram Alpha), I woluld like if it ...
1
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0answers
28 views

Change of variables in a double integral - how to find the region??

I have the following question: Calculate the double integral $\int \int _D (x^4-y^4) e^{xy} dx dy $ on the region $D$, which is the set located in the first quadrant, bounded by the hyperbolas $xy=...
2
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0answers
58 views

Integrating sine with Monte Carlo / Metropolis algorithm

I'm learning Monte Carlo / Metropolis algorithm, so I made up a simple question and write some code to see if I really understand it. The question is simple: integrating sine over 0 to PI. The ...
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0answers
41 views

Is there a product integral that preserves zeroes?

The integral essentially takes the arithmetic mean of the range of a function multiplied by the domain, adding together each possible output weighted by the amount of the domain accounted for by that ...
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0answers
22 views

Backwards Euler for gravitational equation

I have a set of ODEs that simulates a body that moves. Let's say a meteor falling towards the sun. Implementing the explicit Euler is easy $\vec{d}_{n+1} = \vec{d}_n + \Delta t\vec{v}_n$ $\vec{v}_{n+...
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2answers
42 views

Calculating the volume bounded between a paraboloid and a plane

Will someone please help me with the following problem? Calculate the volume bounded between $z=x^2+y^2$ and $z=2x+3y+1$. As far as I understand, I need to switch to cylindrical coordinates: $(...
2
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2answers
57 views

Improper integral involving trigonometric function

I was wondering what happens when evaluating an improper integral involving a trigonometric function where the denominator is a rational function with a zero at $x=0$. The example I have in mind is $...
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4answers
62 views

Integration with variable in numerator

$$\int\frac{x^5}{\sqrt{25-x^2}}dx$$ I tried to do it with substitution but couldn't get ride of $x^5$ in the numerator.
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2answers
90 views

integral problem - what is the quickest solution?

I am solving the below integral. $$\int_{0}^{1} (e^{\frac{-x}{a}}-a(1-e^{-\frac{1}{a}}))^2 dx$$ I can decompose the integrand to the simple elements doing all the algebra and then split and ...
2
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1answer
49 views

Trigonometric contour integral

I cannot figure out what I'm doing wrong: $$\int_0^{2\pi} \frac{1}{a+b\sin\theta} d\theta\quad a>b>0$$ $$\int_{|z|=1} \frac{1}{a+\frac{b}{2i}(z-z^{-1})} \frac{dz}{iz}$$ $$\int_{|z|=1} \frac{...
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1answer
45 views

$\int \delta(x + xy/u - a)\delta(y + xy/v - b)f(x,y)dxdy$?

I need help evaluating the following integral: $$\int \delta(x + uxy - a)\delta(y + vxy - b)p(x,y)dxdy$$ where $\delta(x)$ is Dirac-delta function, and $p(x,y)$ is some sufficiently well behaved ...
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0answers
26 views

Can somebody check whether I have calculated this contour integral correctly?

$$\int_{|z-\frac{1}{2}|=1}\frac{e^{-iz}}{z(z-1)(z^2-1)} dz$$ I used the Residue Theorem and got this answer: $2i\pi-\pi e^i -\frac{3}{2}\pi i e^i$ Is there also some software that can compute these ...
9
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2answers
220 views

Integral $\int_0^\infty\Big[\log\left(1+x^2\right)-\psi\left(1+x^2\right)\Big]dx$

I found this intriguing integral: $$\int_0^\infty\Big[\log\left(1+x^2\right)-\psi\left(1+x^2\right)\Big]dx\approx0.84767315533332877726581...$$ where $\psi(z)=\partial_z\log\Gamma(z)$ is the digamma. ...
1
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1answer
40 views

Integral (log and exp)

Question (Someone asked me for help on this integral and I couldn't figure it out myself.) $$ \int_{-∞}^∞ log(1+ae^{-t^2})dt $$ Even taking the Taylor series such that $log(1+ae^{-t^2})$ ~ $...
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1answer
43 views

Improper integrals and convergence

Let C= {(x,y) such that x>0, y> 0}. Let f(x,y) = $\frac{1}{(x^2 +\sqrt x )(y^2 + \sqrt y)}$ Show that $\int_C{f}$ exists, do not attempt to calculate it. Attempt at at solution: I was thinking that ...
1
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3answers
84 views

Is $f(x)=\frac{1}{q}$ for $x=\frac{p}{q}$ and $f(x)=0$ else integrable?

Let $f:[0,1]\to\mathbb{R}$ be defined as: $$ f(x)=\frac{1}{q}\space\text{ if }\space x=\frac{p}{q}\space \text{ for some }\space p,q\in\mathbb{N} \space \text{ with } (p,q)=1\\ f(x)=0 \space\text{ ...
3
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3answers
78 views

Beginner Integration (Substitution)

I am pretty new to calculus and would like a nudge in the right direction in order to complete this question properly (Maybe also correct any misrepresentations I have about integration) So the ...
2
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3answers
143 views

How to integrate $\int (\tan x)^{1/ 6} \,\text{d}x$? [closed]

How do I compute the following integral $$ I=\int (\tan x)^{1/ 6} \,\text{d}x $$
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votes
2answers
44 views

Evaluate$\int_0^{\infty}x^4e^{-3x}\; dx$ using the change of variable: $t=3x$

I have been given the hint that I need to use integration by parts more than once in order to get the answer. However, I can't seem to get a reasonable result.
1
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2answers
61 views

Evaluating the integral $\int\frac{x^{1/6}-1}{x^{2/3}-x^{1/2}}dx$.

How can I evaluate the following integral $$\int\frac{x^{1/6}-1}{x^{2/3}-x^{1/2}}dx.$$
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3answers
92 views

How does the integral $\int_0^{\infty} \ln\left( 1+\frac{1}{x^2} \right) \,\text{d}x$ converge?

I tried using the fact that $\ln(f(x)) < f(x)$ but that doesn't seem to work. It's an improper integral. $$ \int_0^{\infty} \ln\left( 1+\frac{1}{x^2} \right) \,\text{d}x $$
3
votes
0answers
75 views

How to integrate exponential and power function?

I am trying to solve the following integral $$\int_{0}^{\infty}e^{-(ax+bx^c)}\,dx ; ~~~a,b,c>0.$$ I tried using partial functions but that didn't lead to anything. Any suggestion?
3
votes
2answers
42 views

Calculate $\lim\limits_{R\rightarrow\infty}\int_0^\pi \cos(R\cos t)dt$ w.o. Bessel function

Im calculating a complex path integral to calculate $\int_0^\infty \frac{\sin x}{x}dx$. I was able to evaluate everything except the arc $\int_0^\pi i~\exp(iR~e^{it})dt$ where $R$ is the radius. I ...
1
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0answers
50 views

Got stuck while integrating $\int x^x dx$ [duplicate]

What is the integration of $$\int x^x dx$$ And how can I understand whether an integration is possible or not? Is there any rule to understand whether a function is integrable or not?
3
votes
3answers
90 views

Why integration constant is real?

OK, we are all taught at school that the undefined integral of a function $f(x)$ is $$\int f(x)\;\text{d}x = F(x) + k$$ where $F'(x) = f(x)$ and $k \in \mathbb R$. But, why $k$ must be real? I know ...
2
votes
1answer
45 views

Evaluating the integral of a $\cos(\theta)$ within the exponential wrt $\theta$

I want to evaluate the following integral $\int^{2\pi}_0 \, d\theta e^{- i k (x - x')\cos{\theta}}$, where all of the variables are real and $i$ is the imaginary unit. The difficulty is the cosine ...
0
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0answers
34 views

Unclear passage in integration involving Gamma functions

I find myself in need of some advice on an integration problem. Let $F(x,\lambda)=\Gamma(x,\lambda)/\Gamma(x)$, $x,\lambda>0$ be the Regularized Upper Incomplete Gamma Function, where $\Gamma(x,\...
3
votes
1answer
28 views

Proving that $x^{\alpha}(1+\Vert x\Vert^{2})^{-k}$ belongs to $L^{2}(\mathbb{R}^{n})$

Let $\alpha\in\mathbb{N}^{n}$ be a multi-index, i.e. $\alpha=(\alpha_{1},\dots,\alpha_{n})$ such that $x^{\alpha}:=\prod_{i=1}^{n}x^{\alpha_{i}}_{i}$. The modulus of a multi-index is defined as the ...
4
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1answer
93 views

How to evaluate this Integral $\int { {\sqrt{5^2+K^2}}dK \over {\sqrt{10^2+K^2}K}} $

While working on an Exact Differential Equation, I encounter the following Integral. $$\int { {\sqrt{5^2+K^2}} \over {K\sqrt{10^2+K^2}}} dK$$ I have tried substitution and all the other elementary ...
1
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1answer
84 views

Can this integral be evaluated/approximated?

I've been trying to evaluate this integral without much success: $\displaystyle \int_{-\infty}^\infty dx\, e^{iax} \frac{1- e^{-c\sinh^2 bx}}{\sinh^2 bx}$ I've tried contour integration. There are no ...
1
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2answers
55 views

Integrals with limits

I am trying to do: $$\int_0^1 x\sin(180x^2)\, dx$$ I use substitution: Let $ u = 180x^2 $ and $ \tfrac{du}{dx}=360x $ $$ \implies du =360x \,dx $$ $$ \implies \frac{1}{360} \, du = dx $$ so we ...
1
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1answer
102 views

Substitution theorem for integrals, a “trick”

In Spivak's calculus, he shows the following example: $\displaystyle \int \dfrac{1+e^x}{1-e^x} dx$ and states that setting $u=e^x $and $du=e^x dx$ would work, even though $e^x$ is not there....yet. ...
3
votes
1answer
61 views

Would the order of Taylor Polynomial change after substitution?

I found the order of Taylor Polynomial is kind of confusing. For example, we know: $$T_4e^x = 1 + x + \frac {x^2} {2!} + \frac {x^3} {3!} + \frac {x^4} {4!}$$ After substitute $x$ as $t^2$, we ...
1
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2answers
69 views

Suppose $f$ is continuous and verifies $f(x) = o(x)$. Does it follow that $\int_{0}^xf(y)\,dy = o(x^2)$?

I know $F(x) = \int_{0}^xf(y) dy$ means $F^\prime(x) = f(x)$, but I have no ideas how to relate it to little-oh test. The only method in my mind is to find example $f(y)$ and $F(x)$, but I don't ...
2
votes
4answers
138 views

How to show $\int_{1}^{\infty} \frac{\sin^2(x)}{x^2}dx$ is finite?

At first, my approach was to directly take the improper integral of it. However, it seems not that easy. Then I tried to find another fraction to make a comparison. I got $\frac{\sin^2(x)}{x^2} <...
1
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3answers
105 views

Integral Problem: $\int_{0}^{1} \sqrt{e^{2x}+e^{-2x}+2}$

I am having trouble with this definite integral problem $$\int_{0}^{1} \sqrt{e^{2x}+e^{-2x}+2} \, dx$$ I know that the solution is $$e - \dfrac{1}{e}$$ I checked the step by step solution from ...
1
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1answer
60 views

Related integral problem to the Gaussian integral

So according to Proving $\int_{0}^{\infty} \mathrm{e}^{-x^2} dx = \dfrac{\sqrt \pi}{2}$, $$\int_0^\infty e^{-x^2}dx=\frac{\sqrt{\pi}}{2}$$ I want to solve for this. $$\int_0^\infty e^{-x^2}\ln(x)dx$$ ...
1
vote
1answer
30 views

Product of Hilbert bases of $L^2(\mathbb{R}^p)$ and $L^2(\mathbb{R}^q)$ is a Hilbert basis for $L^2(\mathbb{R}^{p+q})$

Let $(\alpha_n)_n$ be a Hilbert basis of $L^2(\mathbb{R}^p)$ and let $(\beta_k)_k$ be a Hilbert basis for $L^2(\mathbb{R}^q)$. I need to show that $(\alpha_n \beta_k)_{(n,k) \in \mathbb{Z}}$ is also a ...
2
votes
2answers
91 views

How do I evaluate $\int_{0}^{\infty} u^{z-1}(e^{iu}-1) \, du$?

I am trying to evaluate the following integral that shows up in this paper http://arxiv.org/pdf/1103.4306v1.pdf $I=\int_{0}^{\infty} u^{z-1}(e^{iu}-1)du= \Gamma(z)e^{\frac{iz\pi}{2}}$ for $-...
1
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0answers
63 views

The time taken for a particle to return to its initial position involving integration of $v(t)$

Here, for part (a), I have solved for $v(t)=0$ and arrived at the answer that $t=6$. However, for Part (b), the mark scheme given says to do it in the following way: I understood why the mark ...
33
votes
4answers
492 views

Integral whose upper limit is the integral itself: $\int_{0}^{\int_{0}^{\ldots}\frac{1}{\sqrt{x}} \ \mathrm{d}x} \frac{1}{\sqrt{x}} \ \mathrm{d}x$

I recently encountered the following definite integral: $$\int_0^{\int_0^\ldots \frac{1}{\sqrt{x}} \ \mathrm{d}x} \frac{1}{\sqrt{x}} \ \mathrm{d}x$$ where "$\ldots$" seems to indicate that the upper ...