Google L.I.A.T.E, but this is less important than seeing why the choice of u and v that works works. Which is about working backwards through the product rule. Just in case a picture helps...

... where (key in spoiler) ...

Spoiler:

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Don't integrate - balloontegrate!

Balloon Calculus; standard integrals, derivatives and methods

Balloon Calculus Drawing with LaTeX and Asymptote!