Determine the expression of antiderivative of f(x)=1/cos^4(x)

The question states that f(x) = 1/ (cos
x)^4.

We have to find the integral of f(x) = 1/ (cos
x)^4.

(cos x)^-4 = (sec x)^4 = (sec x)^2 * (sec
x)^2

=> ((tan x)^2 + 1)* (sec
x)^2

=> ((tan x)^2*(sec x)^2+ (sec
x)^2)

Int [ ((tan x)^2*(sec x)^2+ (sec
x)^2)]

=> Int [ ((tan x)^2*(sec x)^2] + Int [ (sec
x)^2)]

We know Int [ (sec x)^2)] = tan
x.

Now to determine Int [ ((tan x)^2*(sec
x)^2]

let  t = tan x , dt / dx = (sec
x)^2

=> (sec x)^2 dx =
dt

So Int [ ((tan x)^2*(sec x)^2
dx]

=> Int [ t^2
dt]

=> t^3 / 3 +
C

=> (tan x )^3 / 3 +
C

So Int [ ((tan x)^2*(sec x)^2] + Int [ (sec
x)^2)]

=> tan x + (tan x )^3 / 3 +
C

The required result is tan x + (tan x )^3 /
3 + C