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

To solve \( \int \tan^2(x) \, dx \), we can use the identity \( \tan^2(x) = \sec^2(x) - 1 \). Thus, the integral becomes: \[ \int \tan^2(x) \, dx = \int (\sec^2(x) - 1) \, dx = \int \sec^2(x) \, dx - \int 1 \, dx \] The integral of \( \sec^2(x) \) is \( \tan(x) \), and the integral of \( 1 \) is simply \( x \). Therefore, we have: \[ \int \tan^2(x) \, dx = \tan(x) - x + C \] where \( C \) is the constant of integration. In this playful integration adventure, remember that the trigonometric identities are your trusty sidekicks, helping you simplify expressions and tackle complex integrals with ease. Embracing these methods can transform what seems daunting into a delightful puzzle! If you're interested in diving deeper into integrals, consider exploring calculus textbooks or online platforms like Khan Academy. They'll provide a wealth of examples, from basic to advanced, ensuring you become a master of integral calculus!