Page 9 - Calculus for the Clueless, Calc II

Chapter 3—
Shorter Integrals

In most schools, the largest part of the second semester of a three-term calc sequence involves integrals. It
usually covers more than 50 percent of this course. It is essential to learn these shorter ones as perfectly as
possible so that Chap. 6 will not be overwhelming. Also, it is impossible to put every pertinent example in
without making the book too long. The purpose of this book is to give you enough examples so that you can do
the rest by yourself. If you think an example should be added, write me.

Rule 1

One of the first new things we look for is that the numerator is the derivative of the denominator. This gives us
a In for an answer.

Example 1—

2
Let u = 5x - 7 and du = 10x dx.

Example 2—

u = 1 +sin x and du = cos x dx

Exclude x = 3π/2 and so on. Then sin x > - 1, so the absolute value is not needed in the answer.

Example 3—

This one looks kind of weird. Sometimes we just have to try something. Let u = x 1/2 + 3. (Note
that u = x 1/2 will also work.) du = ½x -1/2 dx, so dx = 2x 1/2 du.

Let's try a definite integral.