Gavin's Calc II Class Clarification: Sep 8

There are two directions to take with this: the one is to try and simplify the integrand, then see what there is in the table, and the other is to look at the table and see what form it makes sense to shoot for. I tend to do the former, but that may be because I'm more familiar with the table.

Let's think about what the table gives us: there are integrands with

1. products of e^ax, sin(bx), cos(cx), and p(x) (a polynomial),
2. powers of sin(x), cos(x) and products of powers of these functions,
3. factored quadratic terms (x-a)(x-b) in the denominators of fractions,
4. plain quadratics x²+a² in the denominators of fractions, and
5. quadratics of the form x²+/-a² in square roots in the numerator or denominator of a fraction.

So if we want to use the table, we have to be able to turn our integral into one of these forms. (Aside: is there any percentage in memorizing these? It's probably worth remembering what they generally look like, if not the exact terms in them.)

What transformations are we going to need to use these? I can think of a couple of different types (this is not an exhaustive list, but probably hits the high points):

1. substitutions to simplify a compound function (for example, x² cos⁵(x³)) before applying a rule,
2. related to this, getting rid of linear terms in a function, e.g., substituting to get rid of the 2x+1 in e^2x+1,
3. rewriting polynomials to get the right quadratic form (usually by completing the square---for example, y² + 4x + 5 = (y+2)² + 1---and then substituting accordingly, and
4. simplifying a rational function (a polynomial divided by another) with degree greater than or equal to the numerator by either substituting or doing long division---see example 8 in section 7.4, for example.

There are, of course, other possibilities.