Find the general solution of \( (\sec x) \frac{d y}{d x}=5 \) A \( \quad y=-5 \cos x+C \) B \( \quad y=-5 \sin x+C \) C \( \quad y=5 \cos x+C \) D \( \quad y=5 \sin x+C \)

To solve the differential equation \( (\sec x) \frac{d y}{d x}=5 \), we start by rewriting it in a more manageable form. Multiply both sides by \( \cos x \) to eliminate the secant: \[ \frac{d y}{d x} = 5 \cos x \] Now, we integrate both sides with respect to \( x \): \[ y = \int 5 \cos x \, dx \] The integral of \( 5 \cos x \) is \( 5 \sin x + C \), where \( C \) is the constant of integration. Thus, we have: \[ y = 5 \sin x + C \] This means that the correct answer is: D \( \quad y=5 \sin x+C \) Let's double-check the integration to confirm it matches the choices provided! 🧙‍♂️✨ To dive even deeper: The solution here is hinged on understanding the properties of the function. The sine function oscillates between -1 and 1, so \( y = 5 \sin x + C \) will also oscillate. It’s a great example of how calculus can illuminate the behavior of periodic functions! Also, integrating with respect to trigonometric functions can sometimes lead to confusion. Remember not to lose track of constant factors when you differentiate or integrate! Always double-check your integration steps!