What is the indefinite integral of the function 3sinx-4tan^2x ?

We'll apply the property of integral to be
additive:

Int [3sin x- 4(tan x)^2]dx = Int 3sin x dx - Int
4(tan x)^2 dx (*)

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

 Int 3sin dx = 3Int sin x dx= -3 cos x + C
(1)

Int 4(tan x)^2 dx = 4Int [(sec x)^2 -
1]dx

4Int [(sec x)^2 - 1]dx = 4Int (sec x)^2 dx - 4Int
dx

4Int [(sec x)^2 - 1]dx = 4 tan x - 4x + C
(2)

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

Int [3sin x- 4(tan x)^2]dx = -3 cos x- 4 tan x + 4x +
C

The anti-derivative of the trigonometric
function 3sin x- 4(tan x)^2 is Int [3sin x- 4(tan x)^2]dx = -3 cos x- 4 tan x + 4x +
C.