Page 22 - Calculus for the Clueless

Chapter 6—
Integration By Parts

As you will see, there is very little theory in this chapter—only hard work.

Integration by parts comes from the product rule for differentials, which is the same as the product rule for
derivatives.

Let u and v be functions of x.

Integrating, we get

What have we done? In the first integral, we have the function u and the differential of v. In the last integral, we
have the differential of u and the function v. By reversing the roles of u and v, we hope to either have a very
easy second integral or be allowed to proceed more easily to an answer.

Example 1—

ax
ax
If a polynomial multiplies e , sin ax, and cos ax, we always let u = polynomial and dv = e dx, sin ax dx, or
cos ax dx. In this example,

Example 2—

We must let u be a polynomial and dv = e dx 4 times!! However, if you observe the pattern, in time you may
3x
be able to do this in your head. Yes, I mean you. Signs alternate, polynomials get the derivative taken, a 3 is
3x
multiplied on the bottom each time, and e multiplies each term.
The answer will be

Note