Just in case a picture helps...

... where the chain rule...

... is wrapped inside the product rule...

... (legs un-crossed version). Straight continuous lines differentiate downwards (integrate up) with respect to x, and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).

Now fill out the rest of the bottom row, and you'll have an exact product-rule derivative on the left-hand side, and you can integrate both sides with respect to t.

The general drift is...

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!