Math Snap

STEP 1

Assumptions
1. We are given the integral $\int (8 \cos(x) + \sec^2(x)) \, dx$.
2. We need to find the antiderivative of the given function.
3. The constant of integration is denoted by $C$.

STEP 2

The integral can be split into two separate integrals:
$$\int (8 \cos(x) + \sec^2(x)) \, dx = \int 8 \cos(x) \, dx + \int \sec^2(x) \, dx$$

STEP 3

Evaluate the first integral $\int 8 \cos(x) \, dx$.
The antiderivative of $\cos(x)$ is $\sin(x)$, so:
$$\int 8 \cos(x) \, dx = 8 \int \cos(x) \, dx = 8 \sin(x)$$

STEP 4

Evaluate the second integral $\int \sec^2(x) \, dx$.
The antiderivative of $\sec^2(x)$ is $\tan(x)$, so:
$$\int \sec^2(x) \, dx = \tan(x)$$

Combine the results of the two integrals and add the constant of integration $C$:
$$\int (8 \cos(x) + \sec^2(x)) \, dx = 8 \sin(x) + \tan(x) + C$$ The antiderivative is $8 \sin(x) + \tan(x) + C$.