I don't know if it works out or not but try x = 3tany
Okay sooo I do all that and then I get to the part where it loops back to my original integral. But since i use integration by parts twice the two negatives make a positive and then you have somthing like:
A = original integral
A = stuff + A
so i clearly did somthing wrong...
Heres my step my step:
At this point its clearly wrong what did I do?
Integration by parts:
Your integral is now:
Use and .
Now you have:
Integration by parts again:
Let and .
Thus and and the integral becomes: .
This becomes: .
Thus and .
Using a table of integrals, we find that .
Plugging back into the equation a few lines up, we have .
Remember though, that since , .
Thus, becomes .
Using the triangle, if the tangent of an angle is , the secant of that angle is . Plugging everything makes that nasty equation become: .
Simplifying that yields: , which is what I think the answer is.
(You think that's ugly, you should see what it looks like in LaTex.)