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

The integral of sec²(x) with respect to x is tan(x) + C. This result stems from the fundamental properties of derivatives and integrals in calculus.

To understand why this is the case, it is essential to recall that the derivative of tan(x) is sec²(x). Therefore, when we integrate sec²(x), we are essentially reversing the differentiation process for the function tan(x). The integral represents a family of functions whose derivatives yield sec²(x), and these functions can differ by a constant term, hence the inclusion of the constant C in the integral result.

This property of differentiation and integration is pivotal in calculus and underpins many results involving trigonometric functions. Recognizing the link between the functions and their derivatives can greatly aid in solving similar integral problems.