# Tagged Questions

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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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
### 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 ...