Question

how do some integrals of sin, cos, tan, etc. when elvaluating at 0 or infinty equal to pi or pi/some number?

Answer 1

mainly tan likes infinity, but that is because tan is a ratio of \[\sin x \div \cos x\]
so in tan(x) if x is pi/2 that means sin(pi/2)/cos(pi/2) which is equal to (1/0). and the limit of 1/0 is infinity. so looking at inverse tan to infinity

tan(pi/2) = infinity
take the inverse of tan
(pi/2) = atan(infinity)

it all depends on the ratios