Ok, I get integration, normally get these right, etc. I'm doing some review before a test over it tonight and I have one problem that I don't understand how the teacher came to. Wolfram Alpha get the same answer and I can even see the steps it took and it still doesn't make any sense.
You can see it here
in a more readable format.
The answer, apparently, is (1/6)sec(3x) +c
I can get it to (1/6)Integrate[tan(3x)] but the antiderivative of tan(x) is not sec(x)
Wolfram and I do this in a bit different order, but it should not matter.
The wolfram alpha steps then use u-du with u=3x and du=3dx (I pull out 1/2 to get sin/cos first, which becomes tan).
THen wolfram alpha does like me and pulls out the 1/2... and gets tan(u)sec(u) using some sorcery that my feeble mind can't comprehend. I, on the other hand, just have tan(u) and of course the 1/6 from pulling out 1/2 and the u-du stuff.
If you want to see the wolfram steps use this link
and paste in the below.
What am I missing? My way is clearly wrong because tan(u) has no antiderivative, but I don't see where this sec(u) is coming from.