Question

find the most general antiderivative or indefinite integral.\\(\int \left(5\sec x\tan x + 3\sec^{2}x\
ight) dx\\)\\(\int \left(5\sec x\tan x + 3\sec^{2}x\
ight) dx = \square\\)

Explanation:

Step1: Split integral into two terms

$$\int (5\sec x \tan x + 3\sec^2 x) dx = \int 5\sec x \tan x dx + \int 3\sec^2 x dx$$

Step2: Factor out constants

$$= 5\int \sec x \tan x dx + 3\int \sec^2 x dx$$

Step3: Apply antiderivative rules

Recall $\int \sec x \tan x dx = \sec x + C_1$ and $\int \sec^2 x dx = \tan x + C_2$:
$$= 5\sec x + 3\tan x + C$$
(Combine constants $C_1, C_2$ into a single arbitrary constant $C$)

Answer:

$5\sec x + 3\tan x + C$ (where $C$ is an arbitrary constant)