We can solve this indefinite integral by integrating each term separately: $\int (\sin x + \sec^2 x) \, dx = \int \sin x \, dx + \int \sec^2 x \, dx$

Using standard trigonometric integration formulas:

1. $\int \sin x \, dx = -\cos x + C_1$ (since the derivative of $-\cos x$ is $\sin x$)
2. $\int \sec^2 x \, dx = \tan x + C_2$ (since the derivative of $\tan x$ is $\sec^2 x$)

Combining these results and introducing a single constant of integration $C$: $\int (\sin x + \sec^2 x) \, dx = -\cos x + \tan x + C$

• Option A is correct.
• Option B is incorrect because $\int \sin x \, dx = -\cos x$, not $\cos x$ (confusing integration with differentiation).
• Option C is incorrect because the integral of $\sec^2 x$ is $+\tan x$, not $-\tan x$.
• Option D has incorrect signs for both terms.