The integral of tan x is given by ∫ tan x dx = ln |sec x| + C or ∫ tan x dx = -ln |cos x| + C. To derive this formula, we can use the substitution method of integration. We substitute u = cos x, du = -sin x dx, and rewrite the integral as ∫ (sin x / cos x) dx = ∫ (1 / cos x) sin x dx = ∫ (1 / u) (-du) = -ln |u| + C = -ln |cos x| + C. Alternatively, we can use the identity tan x = sin x / cos x and rewrite the integral as ∫ (sin x / cos x) dx = ∫ tan x dx.