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So are there any good example where this trick gives you answers to otherwise very hard problems?

I don't know if this counts, but I solved [problem A5 on the 2005 Putnam exam](https://kskedlaya.org/putnam-archive/2005.pdf) using this method.

So what parameter did you introduce?

The academic oeuvre of Dr Feynman (Nobel laureate) himself, largely.

His academic work is not based on/essentially a clever integral trick.

Multiple clever integral tricks...