It follows from the previous exercise that

and

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

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

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):

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

Since we get

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.)

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