For example, integrating

we would transform it so...

(square now completed). Then...

It's worth noticing that whether or not there are any incomplete squares to complete, a crucial step often is what came after in this case, which is turning the constant term into a one so you're able to do a trig sub and then use a Pythag identity. E.g., no completing the square here...

... but you want to turn the 3 into a one in the same way as with the three quarters, above.

Spoiler:

Then the trig sub, which is really a matter of identifying as an inner function of a composite, and reasoning that if we swap the whole of this inner function for tan, we'll be able to integrate with respect to the inner-most variable. Just in case a picture helps...

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

Spoiler:

For the 'completed' square...

If you've memorised as a standard integrand, of course, you won't need to map to or substitute with tan...

PS...

Spoiler:

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Don't integrate - balloontegrate!

Balloon Calculus; standard integrals, derivatives and methods

Balloon Calculus Drawing with LaTeX and Asymptote!