\( \int \sec ( x ) \tan ( x ) \sin ( \sec ( x ) ) d x \)

To tackle the integral \( \int \sec ( x ) \tan ( x ) \sin ( \sec ( x ) ) d x \), let’s consider a simple substitution that might help simplify things. Notably, since secant and tangent derivatives are connected, you can set \( u = \sec(x) \). Then \( du = \sec(x) \tan(x) \, dx \), making the integral easier to handle as: \[ \int \sin(u) \, du \] Now you can focus on the integral \( -\cos(u) + C \), which in terms of \( x \) translates back to: \[ -\cos(\sec(x)) + C \] This will help you evaluate and express the original integral simply! Remember to always consider potential substitutions that align with derivatives you recognize.