I was looking at how the secant function is integrated. The process is not obvious, and I don't expect it to be but I wanted to know if anyone knows who figured this out. Here's what I'm talking about:
$\begin{align*}\int \sec(x)dx &=\int \sec(x)\cdot \frac{\sec(x)+\tan(x)}{\sec(x)+\tan(x)}dx\\ &=\int \frac{\sec^2(x)+\tan(x)\sec(x)}{\sec(x)+\tan(x)}dx. \end{align*}$
If $f(x) = \frac{1}{x}$, $g(x)=\sec(x)+\tan(x)$, g'(x)=\sec^2(x)+\tan(x)\sec(x)
Then \int \sec(x)dx = \int f(g(x))\cdot g'(x)dx=\int \frac{1}{u}du, where $u=g(x)$
$=\ln|\sec(x)+\tan(x)|+c$
So my question is, who first realised how to do this? Who figured out step 2? It's clever and not that obvious.
(and my sub-question is: why does Arturo insist on re-formatting my questions so that my first statements are centre aligned?! :)