What are the most useful math tricks?

Understand the purpose and soul of what is meant by "integration by substitution". Many people (students particularly who think they're "studying") compute like zombies with no regards to the beauty of what is occurring. What do I mean?

When "substituting", we are essentially transforming the domain of the previous equation that might seem quite tedious to approach analytically in the first place to a range that is much easier on the eyes, the objective: To transform the domain of an integrable function such that we would be able to find its corresponding transformed domain to be much easier on the eyes so much so that we might even find an integration formula for it from your typical table of elementary integrals.


Say I'm a moron (okay), and I seem to be unable to attack analytically the problem of the Reinmann sum of y = sin(2x), alright, so instead, let's transform it from (in indefinite integral form by any constant C implicit in its definition) ∫[sin(2x)]dx = (1/2)∫[sin(u)]du, now, we just have to evaluate sin(u) with respect to the subintervals du, noticing that, this was trasnformed by means of du/dx = 2, being, dx = (1/2)du.

Where, for:

/r/math Thread