Solve the anti-derivative of trigonometric function 8cos x+2tan^2x

To determine the anti-derivative of the given function,
we'll have to evaluate the indefinite integral of 8cos x+2(tan
x)^2.

Int [8cos x+2(tan x)^2]dx = Int 8cos x dx + Int 2(tan
x)^2 dx (*)

We'll solve the first integral from the right
side:

 Int 8cos x dx = 8Int cos x dx= 8 sin x + C
(1)

Int 2(tan x)^2 dx = 2Int [(sec x)^2 -
1]dx

2Int [(sec x)^2 - 1]dx = 2Int (sec x)^2 dx - 2Int
dx

2Int [(sec x)^2 - 1]dx = 2 tan x - 2x + C
(2)

We'll substitute (1) and (2) in
(*):

Int [8cos x+2(tan x)^2]dx = 8 sin x + 2 tan x - 2x +
C

The anti-derivative of the trigonometric
function 8cos x+2(tan x)^2 is Int [8cos x+2(tan x)^2]dx = 8 sin x + 2 tan x - 2x +
C.