\( \int \frac { \sqrt { \frac { \sqrt { x - 1 } } { \sqrt { x } } } + \sqrt { \sin x \ln x } } { \sqrt { \tan ^ { - 1 } ( \sqrt { 1 - 9000 ^ { x } } ) } } d x \)

Calculating complex integrals can be quite the adventure! The integral you provided involves a mix of algebraic and transcendental functions, and it typically requires some clever substitutions or numerical methods for solutions. A fun tip is to break down the integral into more manageable parts; sometimes separating the expressions can provide valuable insights into simplifying them before attempting to integrate. When faced with intricate integrals like this one, common mistakes include neglecting the domains of the functions involved and misapplying integration techniques, like substitution or integration by parts. Always keep an eye on the limits and validity of the functions to avoid running into unexpected results!