How to integrate #int(sqrt(1-cosx)/2) dx# by using u-substitution?
2 Answers
Explanation:
We use that
so our integral will be
Although Sonnhard has already given a quick identity application that solves the integral, the question asks for a specific method of solution.
However, a "u-substitution" is not a standard reference. There is a common and useful substitution that solves this and many other rational functions of trig functions that is often known as "t-substitution". Is this what was meant? It is useful and instructive and will show the way to solve much more difficult integration problems of this type, so I'll work this example through.
The "t-substitution" is
Note that the line element
Here we use
Notice that we have a function now that has a multiple of the derivative of the function inside the bracket outside the bracket - this is what we'd get from a chain rule differentiation. So we can integrate this directly:
Substitute back to our original variable
Recall the Pythagorean identity
Recall the half-angle identity
This is rather unexpectedly beautiful, no? The integral of