Integrals of the form [f'(x)/f(x)]dx

∫[f'(x)/f(x)]dx = log[f(x)]

Using the formula

∫tan x dx = ∫(sin x/cos x)dx

If f(x) = t, f'(x)dx = dt
cos x = t;
-sin x dx = dt
sin x dx = -dt

∫(sin x/cos x)dx = ∫-dt/t = -log |t|+c = - log|cos x|+C
= log |sec x|+C