Intereting Posts

Neither provable nor disprovable theorem
Show that the standard integral: $\int_{0}^{\infty} x^4\mathrm{e}^{-\alpha x^2}\mathrm dx =\frac{3}{8}{\left(\frac{\pi}{\alpha^5}\right)}^\frac{1}{2}$
Calculus and Category theory
A Binet-like integral $\int_{0}^{1} \left(\frac{1}{\ln x} + \frac{1}{1-x} -\frac{1}{2} \right) \frac{x^s }{1-x}\mathrm{d}x$
What is the intuition behind the proof of Abel-Ruffini theorem in abstract algebra?
Summation of a series help: $\sum \frac{n-1}{n!}$
Is there anything special about this matrix?
Primal and dual solution to linear programming
Solving $2^x \equiv x \pmod {11}$
any pattern here ? (revised 2)
Proof $ \int_0^\infty \frac{\cos(2\pi x^2)}{\cosh^2(\pi x)}dx=\frac 14$?
How many Jordan normal forms are there for this characteristic polynomial?
What is the definition of $2.5!$? (2.5 factorial)
Should I be worried that I am doing well in analysis and not well in algebra?
Conjugacy classes in group extensions

For example: $$\int_0^1(15-x)^2(\text{d}x)^2$$

- Limit of factorial function: $\lim\limits_{n\to\infty}\frac{n^n}{n!}.$
- Finding $\lim \limits_{x \to 0} \frac{1 - \cos x}{x}$, given $\lim \limits_{x \to 0} \frac{\sin x}{x} = 1$
- Prove $\sin a=\int_{-\infty}^{\infty}\cos(ax^2)\frac{\sinh(2ax)}{\sinh(\pi x)} \operatorname dx$
- Is there a way to parameterize a path on a sphere?
- Show bounded and convex function on $\mathbb R$ is constant
- How do I find $\lim_{n\to\infty}(\frac{n-1}{n})^n$
- Accurate identities related to $\sum\limits_{n=0}^{\infty}\frac{(2n)!}{(n!)^3}x^n$ and $\sum\limits_{n=0}^{\infty}\frac{(2n)!}{(n!)^4}x^n$
- Equivalent Cauchy sequences.
- Simplifying $\frac{\partial V}{\partial T} \cdot \frac{\partial T}{\partial P} \cdot \frac{\partial P}{\partial V}$
- Help minimizing function

There’s an old joke. A mathematician, a physicist and a engineer are asked by a student what the meaning of $$\int \frac{1}{dx}$$ is.

The mathematician says it is meaningless.

The physicist ponders it for a moment and wonders if there is some way to give it meaning.

The engineer says, “Hmmmm, I used to know how to do that.”

This is a misuse of notation – $(dx)^2$ is essentially meaningless, because $dx$ is not something numeric, it is rather an indication of how we are measuring “area” in the integral.

If you replaced $(dx)^2$ with $d(x^2)$, there would be a meaning we could apply.

Just guessing, but maybe this came from $\frac {d^2y}{dx^2}=(15-x)^2$ The right way to see this is $\frac d{dx}\frac {dy}{dx}=(15-x)^2$. Then we can integrate both sides with respect to $x$, getting $\frac {dy}{dx}=\int (15-x)^2 dx=\int (225-30x+x^2)dx=C_1+225x-15x^2+\frac 13x^3$ and can integrate again to get $y=C_2+C_1x+\frac 12 225x^2-5x^3+\frac 1{12}x^4$ which can be evaluated at $0$ and $1$, but we need a value for $C_1$ to get a specific answer.

As I typed this I got haunted by the squares on both sides and worry that somehow it involves $\frac {dy}{dx}=15-x$, which is easy to solve.

- Some proofs for beginners?
- $p^2$ misses 2 primitive roots
- How do I calculate the odds of a given set of dice results occurring before another given set?
- How to prove that either $2^{500} + 15$ or $2^{500} + 16$ isn't a perfect square?
- Prove that a group generated by two elements of order $2$, $x$ and $y$, is isomorphic to $D_{2n}$, where $n = |xy|.$
- Prove that $\sum\limits_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$?
- What is the best way to solve discrete divide and conquer recurrences?
- Limit of a complex number over its conjugate, as z approach the infinity.
- Show that $1^{p-1} + 2^{p-1} +\ldots + (p-1)^{p-1} \equiv -1 \mod p$
- Why does the Gram-Schmidt procedure divide by 0 on a linearly dependent lists of vectors?
- Prove that bitstrings with 1/0-ratio different from 50/50 are compressable
- Decimal Fibonacci Number?
- Friedrichs's inequality?
- Frequency distribution for N balls in m urns (distributed equiprobably)
- Finitely Presented is Preserved by Extension