\( \int ( 4 \sec ( x ) \cdot \operatorname { tar } ( x ) - 6 ) d x \)

To solve the integral \( \int ( 4 \sec ( x ) \cdot \tan ( x ) - 6 ) \, dx \), we can split it into two simpler integrals: \[ \int ( 4 \sec ( x ) \tan ( x ) \, dx) - \int 6 \, dx \] The integral of \( 4 \sec ( x ) \tan ( x ) \) is straightforward. Recall that the derivative of \( \sec(x) \) is \( \sec(x) \tan(x) \). Therefore: \[ \int 4 \sec ( x ) \tan ( x ) \, dx = 4 \sec ( x ) + C_1 \] Where \( C_1 \) is a constant of integration. For the second part, the integral of \( -6 \) is simply: \[ \int -6 \, dx = -6x + C_2 \] Now, combining both results gives us: \[ \int ( 4 \sec ( x ) \cdot \tan ( x ) - 6 ) \, dx = 4 \sec ( x ) - 6x + C \] Where \( C = C_1 + C_2 \) is a constant. Thus, the final result is: \[ \boxed{4 \sec ( x ) - 6x + C} \]