Find the indefinite integral of the function y = tan^4x + tan^2x.

The indefinite integral of the given function is
written:

Int [(tan x)^4 + (tan
x)^2]dx

We'll factorize by (tan
x)^2:

Int (tan x)^2*[(tan x)^2 +
1]dx

We'll solve the integral using substitution
technique:

tan x = t

We'll
differentiate both sides:

dx/(cos x)^2 =
dt

We'll write the fundamental formula of trigonometry and
we'll get:

(sin x)^2 + (cos x)^2 
=1

We'll divide the relation by (cos
x)^2:

(tan x)^2 + 1 = 1/(cos
x)^2

We'll re-write the integral, substituting (tan x)^2 +
1 by 1/(cos x)^2:

Int (tan x)^2*[(tan x)^2 + 1]dx  = Int
(tan x)^2*dx/(cos x)^2

Now, we'll re-write the
integral replacing the variable x by t:

Int (tan
x)^2*dx/(cos x)^2 = Int t^2*dt

Int t^2*dt = t^3/3 +
C

We'll substitute t by tan x  and we'll get the result of
the indefinite integral of the function:

Int
[(tan x)^4 + (tan x)^2]dx = (tan x)^3/3 + C