Page 209 - Calculus Workbook For Dummies

Chapter 11

Integration Rules for Calculus

Connoisseurs

In This Chapter

Imbibing integration
Transfixing on trigonometric integrals
Partaking of partial fractions

n this chapter, you work on some complex and challenging integration techniques. The
Imethods may seem quite difficult at first, but, like with anything, they’re not that bad at
all after some practice.

Integration by Parts: Here’s How u du It

Integration by parts is the counterpart of the product rule for differentiation (see Chapter 6)
because the integrand in question is the product of two functions (usually). Here’s the
method in a nutshell. You split up the two functions in the integrand, differentiate one, inte-
grate the other, then apply the integration-by-parts formula. This process converts the origi-
nal integrand — which you can’t integrate — into an integrand you can integrate. Clear as
mud, right? You’ll catch on to the technique real quick if you use the following LIATE
acronym and the box method in the example. First, here’s the formula:
For integration by parts, here’s what u du: # udv uv - # vdu.
=

Don’t try to understand that until you work through an example problem. Your first chal-
lenge in an integration by parts problem is to decide what function in your original integrand
will play the role of the u in the formula. Here’s how you do it.

To select your u function, just go down this list; the first function type from this list that’s in
your integrand is your u. Here’s a great acronym to help you pick your u function: LIATE, for
Logarithmic (like lnx)

Inverse trigonometric (like arcsin x)
Algebraic (like x4 3 - 10)
Trigonometric (like sinx)
x
Exponential (like 5 )
I wish I could take credit for this acronym, but credit goes to Herbert Kasube (see his article in
American Mathematical Monthly 90, 1983). I can, however, take credit for the following brilliant
mnemonic devise to help you remember the acronym: Lilliputians In Africa Tackle Elephants.