How do I find the correct step-by-step method to solve a complex definite integral using integration by parts?

I am currently working on calculus problems and getting stuck on a specific definite integral. Every time I apply the integration by parts formula ($u\, dv = uv - \int v\, du$), I seem to get lost in the algebraic simplification or end up with an even more complicated integral than what I started with.Could someone break down the best strategy for choosing $u$ and $dv$ (like the LIATE rule), and show a clear, step-by-step example of how to evaluate it from bounds $a$ to $b$? Any tips on avoiding common sign errors during this process would also be highly appreciated!