Page 23 - Calculus for the Clueless, Calc II

If we have and f(x) is a polynomial and g(x) is e , sin kx, or cos kx, we let u be a polynomial and
kx
dv = g(x), and we integrate by parts, of course.

Next we will consider integrating the arc sin, arc tan, and in. If you had never seen them before, you probably
would never guess that all are done by integration by parts, since there appears to be only one function.

However, mathematicians, being clever little devils, invented a second function so that all three of these
integrals are rather easily done.

Example 3—

Note 1

-1
-1
If we have polynomial or is not there (= 1), and g(x) = ln x or sin x or tan x or sec x,
-1
we let dv be a polynomial or i and u = g(x). Integrate by parts.
Note 2

Although Example 3 is relatively short, some of these are verrrry long and use techniques we will learn later in
this chapter.

We will now do a more complicated problem, e cos 3x dx. Based on what we did before, we can take either
5x
function as u and the rest as dv. It turns out both will work. However, the problem is not quite so easy, as we
will see. Being a glutton for punishment, I will show that the problem can be done two ways.