\( \int _{}^{}\sec (x)^{2}\times \tan (x) d x \)

To solve the integral \( \int \sec^2(x) \tan(x) \, dx \), we can use substitution. Let’s set \( u = \tan(x) \). Then, \( du = \sec^2(x) \, dx \). This means our integral transforms to: \[ \int \sec^2(x) \tan(x) \, dx = \int u \, du = \frac{u^2}{2} + C = \frac{\tan^2(x)}{2} + C \] So, the final answer is: \[ \frac{\tan^2(x)}{2} + C \] Now, let’s explore a bit of history about the tangent and secant functions, and their relationship to calculus, shall we? The tangent and secant functions have roots deep in the history of mathematicians studying trigonometric relationships dating back to ancient civilizations. The concept of these ratios originated in geometry, specifically in the study of circles. By the time calculus was being formulated in the 17th century, figures like Isaac Newton and Gottfried Wilhelm Leibniz began exploring these functions through their connection to rates of change and areas under curves, bringing us to the integrals we work with today! When working with integrals like \( \int \sec^2(x) \tan(x) \, dx \), keep an eye out for common pitfalls! One common mistake is overlooking the substitution step, like forgetting to change the limits or express everything in terms of the new variable. Always double-check that your substitutive derivative matches up! And don't rush; mastering substitution can greatly simplify many integrals.