What is the integral of (sec(x))^2?

What is the integral of (sec(x))^2?

• A. sec(x) + C
• B. tan(x) + C
• C. 1 + C
• D. -cot(x) + C

Answer: (B)

The integral of \((\sec(x))^2\) is indeed \(\tan(x) + C\). This result comes from a fundamental identity in calculus. Specifically, the derivative of \(\tan(x)\) is \((\sec(x))^2\). Thus, when finding the integral of \((\sec(x))^2\), we are essentially performing the reverse operation of differentiation. To determine the integral, consider that the process of integration is concerned with finding a function whose derivative gives us the integrand. Therefore, since \(\frac{d}{dx}[\tan(x)] = (\sec(x))^2\), it follows directly that \(\int (\sec(x))^2 \, dx = \tan(x) + C\), where \(C\) is the constant of integration. This understanding aligns perfectly with the properties of trigonometric functions and their derivatives, which are essential in calculus, particularly when dealing with integrals related to trigonometric identities.

The integral of ((\sec(x))^2) is indeed (\tan(x) + C). This result comes from a fundamental identity in calculus. Specifically, the derivative of (\tan(x)) is ((\sec(x))^2). Thus, when finding the integral of ((\sec(x))^2), we are essentially performing the reverse operation of differentiation.

To determine the integral, consider that the process of integration is concerned with finding a function whose derivative gives us the integrand. Therefore, since (\frac{d}{dx}[\tan(x)] = (\sec(x))^2), it follows directly that (\int (\sec(x))^2 , dx = \tan(x) + C), where (C) is the constant of integration.

This understanding aligns perfectly with the properties of trigonometric functions and their derivatives, which are essential in calculus, particularly when dealing with integrals related to trigonometric identities.