Antiderivatives.What is the antiderivative of f(x)=tan^2x+tan^4x?

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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To determine the antiderivative of the given function , we'll have to  calculate the indefinite integral of the given function:

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

We notice that we can factorize by (tan x)^2 and the integrand will become:

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

We'll solve the integral using substitution technique. We'll note tan x = t:

We'll differentiate both sides:

dx/(cos x)^2 = dt

We'll use 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 antiderivative of the function:

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