Find the most general antiderivative of f(x)=–8e^x–6secant^2(x) where -pi/2<x<pi/2

Note: Any arbitrary constants used must be an upper-case "C".

F(x)=?

you have

∫-8e^x - 6sec^2 x dx
= -8e^x - 6tan x + C

To find the most general antiderivative of the given function f(x) = -8e^x - 6sec^2(x) where -π/2 < x < π/2, you can use the fundamental theorem of calculus and apply basic integration techniques.

Step 1: Find the antiderivative of -8e^x.
The antiderivative of -8e^x can be found by using the power rule for integration, which states that the antiderivative of e^x is e^x itself. However, you also need to consider the coefficient of -8, so the antiderivative becomes:

∫ -8e^x dx = -8∫ e^x dx = -8e^x + C1

Step 2: Find the antiderivative of -6sec^2(x).
The antiderivative of -6sec^2(x) can be found by using the formula for integrating sec^2(x), which is tan(x). However, since there is a coefficient of -6, the antiderivative becomes:

∫ -6sec^2(x) dx = -6∫ sec^2(x) dx = -6tan(x) + C2

Step 3: Combine the results from Steps 1 and 2.
Now, combine the antiderivatives obtained in Steps 1 and 2 to find the most general antiderivative of f(x):

F(x) = -8e^x - 6tan(x) + C

where C = C1 + C2 is the arbitrary constant.

Therefore, the most general antiderivative of f(x) = -8e^x - 6sec^2(x) where -π/2 < x < π/2 is F(x) = -8e^x - 6tan(x) + C, where C is an arbitrary constant.