Next: 17.3 Integration by Parts Up: 17. Antidifferentiation Techniques Previous: 17.1 The Antidifferentiation Problem   Index

# 17.2 Basic Formulas

Every differentiation formula gives rise to an antidifferentiation formula. We review here a list of formulas that you should know. In each case you should verify the formula by differentiating the right side. You should know these formulas backward and forward.

17.7   Exercise. Verify that

and

It follows from the previous exercise that

and

You should add these two formulas to the list of antiderivatives to be memorized.

17.8   Theorem (Sum rule for antiderivatives) If and are functions that have antiderivatives on some interval , and if then , and have antiderivatives on and

and
 (17.9)

Proof:      The meaning of this statement is that if is an antiderivative for and is an antiderivative for , then is an antiderivative for , and is an antiderivative for . The warning about ambiguous notation for indefinite integrals given in chapter 9 applies also to antiderivatives.

Let be antiderivatives for and respectively on . Then and are continuous on , and

on . Hence are continuous on , and

on , and hence

Also is continuous on , and

on , so that

17.10   Example. We will calculate .

I will try to bring this integral into the form

It appears reasonable to take , and then . The doesn't quite appear in the integral, but I can get it where I need it by multiplying by a constant, and using (17.9):

17.11   Example. We will calculate .

This problem is more complicated than the last one. Here I still want to take , but I cannot get the '' that I need. I will multiply out

17.12   Example. We will calculate .

Since we get

17.13   Example. We will consider .

Although this problem looks similar to the one we just did, it can be shown that no function built up from the functions we have studied by algebraic operations is an antiderivative for . So we will not find the desired antiderivative. (But by the fundamental theorem of the calculus we know that has an antiderivative.)

17.14   Example. We will calculate .

17.15   Example. We will calculate .

The integrand looks enough like that I will try to get an from this integral.

Now , so

Thus we have found an antiderivative for . Hence

17.16   Exercise. A Find the following antiderivatives:
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)

Next: 17.3 Integration by Parts Up: 17. Antidifferentiation Techniques Previous: 17.1 The Antidifferentiation Problem   Index
Ray Mayer 2007-09-07