The document calculates the indefinite integral of the function y = tan^4x + tan^2x. It uses u-substitution, where tan x = t, and trigonometric identities to rewrite the integral in terms of t. It then evaluates the integral of t^2 dt to get the final solution of (tan x)^3/3 + C.

1. Calculate the indefinite integral of the function
y = tan^4x + tan^2x.
Solution
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