For K_n = integral tan^n(x) dx: Write tan^n(x) = tan^(n-2)(x) * tan^2(x) = tan^(n-2)(x)(sec^2(x)-1). K_n = integral tan^(n-2)(x)sec^2(x)dx - K_(n-2) = tan^(n-1)(x)/(n-1) - K_(n-2)

For L_n = integral sec^n(x) dx: Use parts: u = sec^(n-2)(x), dv = sec^2(x)dx. Result: L_n = sec^(n-2)(x)tan(x)/(n-1) + (n-2)L_(n-2)/(n-1)

Base cases:

• K_0 = x, K_1 = -ln|cos x| = ln|sec x|
• K_2 = tan(x) - x
• L_0 = x, L_1 = ln|sec x + tan x|
• L_2 = tan(x)
• L_3 = (sec(x)tan(x) + ln|sec x + tan x|)/2

Key observation: tan and sec reduction formulas reduce by 2, like sin and cos.