What did I do wrong?
The correct answer is
you set up the triangle after you see what equation you should use. you should have seen that and set up the triangle based on that. then you wouldn't have this business. be sure to review trig sub so you know what equation to start with and hence how to set up the right triangle (no pun intended, well, maybe a little )
Haha, i know. i hate trig sub as well. This is what i thought of when i first saw the integral (then i sighed, grit my teeth and did what the OP asked for)
Let and the integral becomes
is there an easier way?
I set up the triangle so that the hypotenuse was , the adjacent angle to theta was , and the opposite angle to theta was . I'm guessing that when the triangle is set up the constant should be placed adjacent to theta when possible in order to make the process of working the problem out easier. Is this correct?
no, you don't decide where the things go. your equation does.
you had the expression
this is of the form . the proper trig sub is
So that is here.
This means that which means that:
Your acute angle is
The opposite side is
The adjacent side is and
The hypotenuse is
The radical expression is not always the hypotenuse, our expression dictated where we put stuff