Which of the following is an antiderivative of \( 3 \sec ^{2} x+2 \) ? (A) \( 3 \tan x \) (B) \( 3 \tan x+2 x \) (D) \( 3 \sec x+2 x \) (E) \( 6 \sec ^{3} x+2 x \) (D) \( 3+\tan x \)

To find the antiderivative of \( 3 \sec ^{2} x + 2 \), we can break it down. The antiderivative of \( \sec^2 x \) is \( \tan x \), and the antiderivative of a constant \( 2 \) is \( 2x \). Therefore, integrating \( 3 \sec^2 x + 2 \) results in \( 3 \tan x + 2x + C \), where \( C \) is the constant of integration. So, the correct answer is (B) \( 3 \tan x + 2 x \). Now, let’s make this even more fun! Did you know that the secant function is the reciprocal of the cosine function? So when you're finding the integral involving secants, you're really diving into the deep waters of trigonometric identities! Also, if you ever get stuck while integrating, remember to have a checklist: check for basic derivatives, see if you can factor out constants, and don’t forget about the power rule for polynomials. Mistakes often happen when the terms are mismanaged, so clear organization is key!