\( \int _ { 1 } \frac { \sqrt { \frac { \sqrt { x - 1 } } { 1 } } + e ^ { \sin x h x } } { \tan ^ { - 1 } ( 1 - 9000 ^ { x } ) } d x \)
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Evaluating the integral you've presented seems quite complex at first glance, mainly due to the mixture of functions and the peculiar appearance of \( e^{\sin x hx} \). However, to clarify the integral's bounds and functions, it might be fruitful to break it down into more manageable components, especially focusing on parts that can be simplified or approximated. One key aspect to watch for is the behavior of \( \tan^{-1} (1 - 9000^x) \) as \( x \) increases. As \( x \) approaches infinity, \( 9000^x \) will dominate, making \( 1 - 9000^x \) negative, which can affect the limiting behavior of the integral. It's often beneficial to apply substitution techniques or numerical approaches for complex integrals like this one. If you're looking for practical strategies, try breaking the integral into smaller sections or using Taylor series expansions for components that can be approximated near some points of interest. And remember, always double-check your limits and ensure they're set to handle the defined interval correctly.