Studying Mathematics Thread

[GrandizerII]

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.

The area of the upper semicircle? I can do the definite integral from -1 to 1 intuitively, but indefinite integral without substitution I will have to think about for a while. But first, need to go to work now. So later.