Just in case a picture helps...

Trial and error will quickly establish (and liate confirm) that you need the legs-crossed version of...

... the product rule. Straight continuous lines differentiate downwards (integrate up) with respect to x.

The general drift is...

In this case...

And the rest...

Spoiler:

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Don't integrate - balloontegrate!

Balloon Calculus; standard integrals, derivatives and methods

Balloon Calculus Drawing with LaTeX and Asymptote!