Neat expressions that equal 1 I would like to see beautiful and elegant expressions involving elementary and non-elementary functions, transcendental numbers, etc. that equal 1. 
Be creative!
 A: Because any proper probability distribution must integrate to one, there can be a host of them. Here are two such:
$$
\large\int_{-\infty}^{\infty}\frac{1}{\pi(1+x^2)}dx\\
\large\int_{0}^{\infty}\left[\frac{\lambda}{2 \pi x^3}\right]^{\frac{1}{2}}e^{\frac{-\lambda (x-\mu)^2}{2 \mu^2 x}}dx\\
$$
A: $$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{\frac{-x^2}{2}}dx$$
Also
$$\pi\int_0^{\infty}e^{-\pi{x}}dx$$
A: Here is one expression: $-e^{i\pi}$
A: The sum of the solutions $x\in\mathbb{C}$ of $$\sum_{n=0}^{2^{32}}(-1)^nx^{2^{32}-n}=0$$
A: $$1 = -\cos\left(2\int_{-1}^1 \!\sqrt {1-x^2}\,dx\right)$$
$$1 = -1+\frac{\pi^2}{2!}-\frac{\pi^4}{4!}+\frac{\pi^6}{6!}-\cdots$$
A: For all positive integers $n$,
$${\sum_{i=0}^n{n\choose i} \over {2^n}}$$
For all polynomials $p$ with leading coefficient $a_p$ and degree $k$,
$${{d^kp \over dx}\over a_p \cdot k!}$$
For all integer polynomials $q$ having leading coefficient $a_q$ and degree $k$,
$${\sum_{i=0}^k{(-1)^i{k\choose i}(q(x-i)-q(x-i-1))}\over a_q\cdot k!}$$
A: $\lim_{n\to\infty}n\log(1+n^{-1})=1$
A: I like using $~7$th-grade expressions :$$15(x / 5) + 15(x - 1) / 3 = 15(1 / 5) $$
A: Let {$x_n$} be a sequence of strictly positive numbers with the following propriety: 
$$ \sum_{k=1}^n k x_k = \prod_{k=1}^n x_k^k $$ for every positive integer $n$.
It turns out that 
$$ \lim_{n \to \infty} x_n = 1 $$
