find the most general antiderivative or indefinite integral. ∫(-9 sec x tan x + 5 sec²x) dx ∫(-9 sec x tan x…

Explanation:

Step1: Use integral - sum rule

$\int(-9\sec x\tan x + 5\sec^{2}x)dx=\int(-9\sec x\tan x)dx+\int(5\sec^{2}x)dx$

Step2: Use constant - multiple rule

$=- 9\int\sec x\tan xdx+5\int\sec^{2}xdx$

Step3: Recall antiderivative formulas

We know that $\int\sec x\tan xdx=\sec x + C_1$ and $\int\sec^{2}xdx=\tan x + C_2$. So, $-9\int\sec x\tan xdx+5\int\sec^{2}xdx=-9\sec x + 5\tan x+C$ (where $C = C_1 + C_2$)

Answer:

$-9\sec x+5\tan x + C$