What are some examples of mathematical facts that had once been open problems for a significant amount of time and thought hard or unsolvable by contemporary methods, but were then unexpectedly solved thanks to some out-of-the-box flash of genius, and the proof is actually short (say, one page or so) and uses elementary mathematics only?

**Answer**

The integral of $\sec x$ stumped mathematicians in the mid-seventeenth century for quite a while until, in a flash of insight, Isaac Barrow showed that the following can be done:

$$\int \sec x \,\mathrm{d}x= \int \frac{1}{\cos x} \, \mathrm{d}x=\int \frac{

\cos x}{\cos^2 x} \, \mathrm{d}x=\int \frac{

\cos x}{1-\sin^2 x} \, \mathrm{d}x.$$

Using $u$-substitution and letting $u=\sin x$, the integral transforms to

$$\int \frac{1}{1-u^2} \, \mathrm{d}u,$$

which is easily evaluated by partial fractions.

**Attribution***Source : Link , Question Author : Damian Reding , Answer Author : Community*