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.

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)