Question 80870: Find an antiderivative of the function
(7 *e^(2 *x ))/(5+e^(4*x))
involving the arctan function.
Note. Powers of e may be input either directly or using the function exp, e.g. e2 may be input as e^2 or exp(2). There is an equivalent solution to the problem in terms of ln (or log). However, the solution is required to be in terms of the arctan function (i.e. the inverse of the tan which is sometimes written as tan-1). Please enter arctan and not tan-1.
Answer by kev82(151)   (Show Source):
You can put this solution on YOUR website! Hi,
A nice substitution will make this entire thing collapse into nothingness. As my old maths teacher used to say many many years ago, integration is all about intelligent guesswork. The correct substitution to make this collapse is:
(If you need me to post the working for this substitution to the final answer then post back, but I'm guessing your problem is actually coming up with the right substitution, not the algebra.)
But without years of practice, it's probably not particularly clear how I saw that. With that subsitution, I'm actually achieving a few things, so I'm going to list them separately, each with their substitutions which will hopefuly give you an idea of how to tackle these problems.
The varibale we're integrating over, only appears as exp(2x) and exp(4x), so we can make a subsitution for the exp, but the question is what should we subsitute for. You can try substituting for exp(x), but it doesn't look particularly nice. I would tend to always try substituting for one of the things I had (either exp(2x) or exp(4x)). The idea being that substituting for something I have will make it go away. You can substitute for y=exp(4x) but then exp(2x)=sqrt(y) and square roots are nasty creatures. So I would go with y=exp(2x).
If you do this you should be left with:
The denominator should be shouting out at you trig! trig! trig!. If factor a 5 out of there we have  or to make the substitution more obvious to you ^2) .
Were looking for a trig identity where ^2=\mbox{something else}) The clue is given to you in the question, because they say arctan, so you're looking for something with a tangent in it. The identity is of course  . So we need  .
Putting this in, the denominator turns into a  and so does the top because of the conversion from dy to dz. They cancel to reveal.
Now, I'm sure you can do that integral. It then just comes down to converting z back to x with the substitutions you've made.
If you need any help with the intermediary algebra then post back, and once you've got the answer, post back again so we can see how you did. As I said at the top, it's all about experience and intelligent guesswork, just keep trying lots of them. If you have a friend who wants to practice with you then each of you could pick a function, differentiate it, and then give it to the other to try and integrate.
Hope that helps,
Kev
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