Antiderivative of a square root

Substitute , so that . The integral is then equivalent to

In general, , so .

We want the substitution made above to be reversible, so that . This restricts to the interval , and over this interval we have , so we take the positive square root in order that .

Then the integral becomes

which can be computed in several ways. One method is to integrate by parts, taking

so that

Then with , you have and , which follows from the Pythagorean identity. So

stops on an even number.