I am supposed to know this by heart and I do:
http://www.wolframalpha.com/input/?i=integral+(1%2F(x^2+%2B+1))
However, on a specific problem I'm having, I need to know to get the following:
http://www.wolframalpha.com/input/?i=integral+1%2F(x^2+%2B+36)
My question is:
Is there a way for me to use what I have memorized to derive the second link? Like how can I make sense of the second link?
Any input would be greatly appreciated!
Thanks in advance!
Eek, its better to just use LaTex (if you know how) and link the problem.
However this isn't a "simple" integral, unless you are familiar with trig substitutions. Lets do a little bit of math trickery:
Thus:
From here we simplify:
This integral is easy to evaluate:
However, we need our final answer in terms of x, not theta. So, we go back to our original substitution:
. . .and solve for theta. Do not "memorize" formulas or "tricks" (this really isn't a trick though I called it so) - memorize (or KNOW) identities, and familiar algebraic structures.
Sorry, I was typing when you posted; your way works well, too.
And s3u: if you're going to take a timed test like AP Calc AB, memorizing could save a lot of time. Of course, you don't need to memorize it, I'm sure, because you saw where that came from, you could re-figure that out in a few seconds.