How do you find the integral of secx and cscx in a step-by-step manner?

Question

Please use calculus.

Answer 1

Whoops...I derived instead of integrated (d'oh).
The integral of secant can be found here: http://calc101.com/special_3.html

The integral of cosecant can be found here:
http://calc101.com/special_4.html

--- derivation ---
Both are derived from sin x and cos x using the quotient rule and their definitions:

f(x) = sec x
sec x = 1 / cos x

Quotient rule: For f(x) = g(x)/h(x), f'(x) = (g'(x)h(x) -g(x)h'(x)) / h^2
g(x) = 1
h(x) = cos x
g'(x) = 0
h'(x) = -sin x
h^2(x) = cos^2 x

f'(x) = 0cos x - 1(-sin x) / cos^2 x
f'(x) = 0 + sin x / cos^2 x
f'(x) = sin x / cos^2 x = sin x / cos x * 1 / cos x = tan x sec x (solution!)

f(x) = csc x
csc x = 1 / sin x

Quotient rule: For f(x) = g(x)/h(x), f'(x) = (g'(x)h(x) -g(x)h'(x)) / h^2
g(x) = 1
h(x) = sin x
g'(x) = 0
h'(x) = cos x
h(x) = sin^2 x

f'(x) = 0sin x - 1(cos x) / sin^2 x
f'(x) = 0 - cos x / sin^2 x
f'(x) = -cos x / sin^2 x = -cos x / sin x * 1 / sin x = -csc x cot x (solution)