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

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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.**