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**Ackbeet** True, although later on down, they mention ILATE as an alternative that many people prefer (I definitely prefer it).

For trig substitutions, I always draw a right triangle, and assign sides based on this rule:

1. If the expression looks like $\displaystyle x^{2}+y^{2},$ then one side is x, and the other y, and the hypotenuse is $\displaystyle \sqrt{x^{2}+y^{2}}.$ Which side is x and which is y doesn't materially matter, though sometimes one way might be easier than the other.

2. If the expression looks like $\displaystyle x^{2}-y^{2},$ then x is the hypotenuse, y is one of the sides, and the other side is $\displaystyle \sqrt{x^{2}-y^{2}}.$ Which side is y doesn't materially matter, though sometimes one way might be easier than the other.

You go through with the integration. If it's a definite integral, I'll often do the whole thing in the angle domain (after transforming the limits, the integrand, and the differential). If it's an indefinite integral, I'll use the triangle again to get back to the original variables.

By-parts corresponds to the product rule for derivatives, substitutions correspond to the chain rule. Incidentally, there is an analogy for the quotient rule:

$\displaystyle \int\frac{fg'-gf'}{f^{2}}\,dx=\frac{g}{f}+C.$

Unfortunately, it's not all that useful, because the exact form there is not all that common.