Evaluate the following integral for .
(Note: This is a pretty involved problem the way I’ve done it. Maybe there’s a better way? Let me know if you have one. Also, there is an error in the answer in the book on this problem and the next one. The answers given in the book are swapped, so the answer listed for this problem #46 is actually the answer for #47 and vice-versa.)
There are some integrals we’ll want to use to carry out the evaluation of the above integral. First, from previous exercises here and here (Section 5.10, Exercises #7 and #10(b)) we know
Therefore, (we’ll want this later), we have
The other integral that we are going to want to have available is
To evaluate this we’ll use the trig integrals above. First, make the substitution
and also gives us
Therefore we have
Then, substituting back in for (and noting that and ) we have
So, now that we have those, we can turn our attention to the integral in the question. For this integral we want to make the substitution
which implies
Therefore we have
Now, we want to make the substitution
and implies
Therefore,
Now, we can use the work we did above in the evaluation of this integral,
Finally, we have to unwind our substitutions to get back to a function of . We have
Therefore,
This completes our evaluation of the integral.
Wow crazy stuff, thanks so much for posting this!!!
Very long method
it’s easier using x-a=(b-a)sin^2(u)
doing a little bit of algebra sqrt((x-a)(b-x))=|a-b|sqrt(1-((2x-a-b)/(a-b))^2)/2
then by substitution u=(2x-a-b)/(a-b) you integrate (|a-b|/2)integral(sqrt(1-u^2))
instead of |a-b|/2 it should be |a-b|/(a-b)