Questions: Evaluate ∫ sec(x) · [2 sec(x) + 12 tan(x)] dx. Let C represent the constant of integration. ∫ sec(x) · [2 sec(x) + 12 tan(x)] dx = □ + C

Transcript text: Evaluate $\int \sec (x) \cdot[2 \sec (x)+12 \tan (x)] \mathrm{d} x$. Let $C$ represent the constant of integration. $\int \sec (x) \cdot[2 \sec (x)+12 \tan (x)] d x=$ $\square$ $+C$

Solution

Solution Steps

To evaluate the integral $\int \sec(x) \cdot [2 \sec(x) + 12 \tan(x)] \, dx$, we can distribute the $\sec(x)$ term inside the brackets and split the integral into two separate integrals. This results in $\int 2 \sec^2(x) \, dx + \int 12 \sec(x) \tan(x) \, dx$. Each of these integrals can be solved using standard integration techniques: the first integral is a standard integral of $\sec^2(x)$, and the second is a standard integral of $\sec(x) \tan(x)$.

Step 1: Distributing the Function

We start with the integral: \[ \int \sec(x) \cdot [2 \sec(x) + 12 \tan(x)] \, dx \] Distributing $\sec(x)$ gives us: \[ \int (2 \sec^2(x) + 12 \sec(x) \tan(x)) \, dx \]

Step 2: Splitting the Integral

We can split the integral into two separate integrals: \[ \int 2 \sec^2(x) \, dx + \int 12 \sec(x) \tan(x) \, dx \]

Step 3: Evaluating Each Integral

The first integral, $\int 2 \sec^2(x) \, dx$, evaluates to: \[ 2 \tan(x) \] The second integral, $\int 12 \sec(x) \tan(x) \, dx$, evaluates to: \[ 12 \sec(x) \]

Step 4: Combining the Results

Combining both results, we have: \[ \int \sec(x) \cdot [2 \sec(x) + 12 \tan(x)] \, dx = 2 \tan(x) + 12 \sec(x) + C \]