Consider the integral ∫arcsin⁡(4x) dx\displaystyle \int{\arcsin{{\left({4}{x}\right)}}}\ {\left.{d}{x}\right.}:

Applying the integration by parts technique, let

u=\displaystyle {u}= and dv=\displaystyle {d}{v}= dx\displaystyle {\left.{d}{x}\right.}
 

     

Then uv−∫vdu\displaystyle {u}{v}-\int{v}{d}{u} = -∫\displaystyle \int dx\displaystyle {\left.{d}{x}\right.}

                                   

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