# Tagged Questions

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### Why do we need the absolute value signs when integrating the square of a function?

Why do we need the absolute value signs in the definition of square-integrable function? As seen below: $$\int_{-\infty}^{\infty} \lvert f(x) \rvert^2 dx < \infty$$
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### Integral notation

I have encountered the following integral: $\int_{x-d}^{x+d}f(y)dy$ I am trying to figure out what is the role of $d$ in this integral. Is the $d$ at the beginning of the integral the same as the ...
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### Why are variables in integration by substitution so counter intuitive?

Integration by substitution is defined as something like $\displaystyle\int_a^b f(\phi(t))\phi'(t)dt = \int_{\phi(a)}^{\phi(b)} f(x)dx$ But for my taste, the variables $x$ and $t$ are exactly ...
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### How do you write an integral and why

A. Year 1 Calculus Student Approach $$F(x) = \int f(x') dx\,$$ B. Random math paper you find online approach $$F(x) = \int dx f(x') \,$$ C. Spivak $$F(x) = \int f(x) \,$$ D. ??? (Edit) ...
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### Why don't we indicate the variable to summed as we do for integrals?

When integrating over a certain variable $x$, we make sure to end the integral with $dx$, like so: $$\int_{1}^{\infty}\frac{1}{x^2}dx$$ The reason for this of course becomes more clear as one goes ...
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### Notation for Surface Integral in $\mathbb{R}^3$

Recently, a paper of mine got accepted, but the reviewers are struggling with the (in my view) standard notation for surface integrals in $\mathbb{R}^3$: Let $\Gamma \subset \mathbb{R}^3$ be a ...
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### Integral notation from cartesian from polar coordinates

Given an integral $$I=\int\limits_{\mathbb{R}^n} \cdot \; dx,$$ we can introduce polar coordinates, such that $$I=\int\limits_{\Bbb S^{n-1}} \cdot \; d\theta.$$ Another way to express the latter one ...
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### Integration notation

could someone explain the following notation: $u(x,t):=\int^t_0 v(x,t:\tau)d \tau$ It's come up but I don't understand how to interpret the semicoloned tau
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### Integrating a term again and again

So if you have $f''(x) = 24x$ you know you want to integrate it, because it would look much better integrated, so now we have $f'(x) = 12x^2$, but it could still look better, so we integrate it to ...
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### Strange Integral Notation? [duplicate]

When reading about an certain algorithm (about parameter estimation for Kalman Filtering page 7 eq 57) I found this notation: $\int dx f(x)$ which is normally written as $\int f(x) dx$. I spent a ...
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### Intuition for Integration of Differential Forms

In mathematics, we define $dx^i$ as linear functionals, when speaking of integration. However, in physics, we interpret $dx^i$ as very small quantities. There is nothing inherently small about a ...
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### Why $dt/t$ in Mellin transform

I've noticed that often when people write the Gamma function $\Gamma(s) = \int_0^\infty t^{s-1}e^{-t}\,dt$, that they write it like $$\Gamma(s) = \int_0^\infty t^s e^{-t}\,\frac{dt}{t} ,$$ where ...
### Why do people break up the derivative notation when setting $u$ and $v$ for integration by parts?
If we take for example, the problem of $$\int e^x \sin x \quad dx$$ We use the integration by parts technique: $$\int uv' = uv - \int vu'$$ Setting \$\begin{array}{l l} u = \sin x & ...