Studying Mathematics Thread

[polymath257]

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

A related integral is that of sqrt(1-x^2). But this one can be done by 'elementary' methods. No substitution.

Hint: interpret as an area.