The most flexible method of integration?
The tool of mathematics is easily considered the most exact of the sciences. However, if these usually very precise answers to homework and exam problems were cities or towns on a map, then part of math's beauty is that several roads can lead into it. We have perhaps seen or worked word problems which could be solved using more than one branch of math, for example, calculating the required dimensions of a tunnel entrance utilizing either geometry or trigonometry.
Calculus is understandably difficult for many, if not most, of us, but its arsenal of differentiation or integration techniques allow it to be quite flexible when its applied to homework assignments or exam problems. For instance, differentiating a polynomial divided by another polynomial can achieved (after any simplifications) using either the quotient rule or the product rule (from which the former is derived anyway).
If differential calculus is primarily protocol, then integral calculus is significantly recognition. And sometimes students have to perform integration on a function that doesn't conveniently lend itself to most, if not all, of the methods they are shown initially. For that we are taught integration-by-parts. Its “proof” is a straightforward derivation from the product rule, a differentiation procedure of all things (udv = uv – vdu). But, it introduces us to a technique of great adaptability and wide application. And as a method within a major branch of calculus, integration-by-parts by itself can provide more than one road into that precise answer.
Here are a few useful integration-by-parts “rules-of-thumb” to consider when facing an integral, especially during an exam where time is at a premium:
1) if you know the derivative of a function, but don't know (or can't quickly recall) its integral (e.g., arctan x or ln x), let that function be “u”;
2) if one of the functions is a polynomial, let that function be “u” unless “rule” # 1 applies;
3) if one of the functions is sin x or cos x, where the 2nd derivative of the function equals minus the function (y'' = – y), let that function be “u” unless “rule” # 1 or # 2 applies.
In fact, years after many of you struggled with undergraduate calculus, Professor Herbert Kasube of Bradley University proposed the “Liate” (L.I.A.T.E.) rule to advise that whichever function comes first in the following list should be “u”:
I (Inverse trigonometric): arctan x, arcsec x, etc.,
A (Algebraic): x2, 3x50, etc.,
T (Trigonometric): sin x, tan x, etc.,
E (Exponential): ex , 19x, etc.
Basically, the function which is to be “dv” comes later in the list, that is, functions lower on the list have easier anti-derivatives than the functions above them.
1. Kasube, Herbert E. (1983). "A Technique for Integration by Parts". The American Mathematical Monthly 90 (3): 210–211.