Neat expressions that equal 1

Question

I would like to see beautiful and elegant expressions involving elementary and non-elementary functions, transcendental numbers, etc. that equal 1.

Be creative!

Answer 1

Because any proper probability distribution must integrate to one, there can be a host of them. Here are two such: $$ \large\int_{-\infty}^{\infty}\frac{1}{\pi(1+x^2)}dx\\ \large\int_{0}^{\infty}\left[\frac{\lambda}{2 \pi x^3}\right]^{\frac{1}{2}}e^{\frac{-\lambda (x-\mu)^2}{2 \mu^2 x}}dx\\ $$

Answer 2

Here is one expression: $-e^{i\pi}$

Answer 3

$$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{\frac{-x^2}{2}}dx$$

Also

$$\pi\int_0^{\infty}e^{-\pi{x}}dx$$

Answer 4

$$1 = -\cos\left(2\int_{-1}^1 \!\sqrt {1-x^2}\,dx\right)$$ $$1 = -1+\frac{\pi^2}{2!}-\frac{\pi^4}{4!}+\frac{\pi^6}{6!}-\cdots$$

Answer 5

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 $k$,

$${\sum_{i=0}^k{(-1)^i{k\choose i}(q(x-i)-q(x-i-1))}\over a_q\cdot k!}$$

Answer 6

The sum of the solutions $x\in\mathbb{C}$ of $$\sum_{n=0}^{2^{32}}(-1)^nx^{2^{32}-n}=0$$