Studying Mathematics Thread

[GrandizerII]

This took quite a bit of effort, but here's how you derive the integral of secx "from scratch". Whoever came up with such solution was pretty damn smart.

Here's the work (thanks to YouTube, of course!):
https://pasteboard.co/HiEZbdk.png

That's the cursor at the end of the last line (not an absolute value bar).

Much easier is to realize that the derivative of

sec x + tan x

is

sec x tan x + sec^2 x = sec x(sec x + tan x).

So, write

int sec x dx = int [ sec x (sec x + tan x) ]/(sec x + tan x) dx

and do a substitution u= sec x + tan x. The result falls out.

Yeah, that seems to be the better way of arriving at the expected answer, but requires thinking about differentiating secx+tanx in the first place. There's a reason I call solving these problems a form of art. Lots of creative thinking needed. No way I can do all this on my own without any assistance (at this stage, at least).

Here's my solution to the integral of sqrt(1+x^2). The triangle method comes in handy here.

https://pasteboard.co/HiNzLeB.png