I am trying to find the integral of (e^8x)/(e^16x)+4?

Answer 1

It sounds like u = e^(8x) is a good substitution.
du = 8*e^(8x)*dx

So you end up with
1/8 * du/(u^2 + 2^2)

That's a standard integral
1/8 * 1/2 * arctan(u/2)
= 1/16 * arctan[e^(8x)/2] + constant

Answer 2

I guess you mean (e^8x)/(e^16x)+4), or it'd be quite trivial. Put t = e^8x, so that e^16x = t^2. It follows x = ln(t)/8, so that dx = t/8 dt. Then, we have to integrate

Int t/(t^2 + 4). (1/8) /t dt = 8 Int dt/(t^2 + 4) Since 4 = 2^2, this integral is well known, it is (1/2) arctan(t/2) + C, so that

Int t/(t^2 + 4). (1/8) /t dt = 4 arctan(t/2) + C

Going back to x, we get

Int (e^8x)/(e^16x)+4) = 4 arctan((e^8x)/2) + C

Answer 3

Is the denominator [(e^16x)+4]?

Try u=e^8x.

What's u^2?

Do you have arctan in any of your integration formulas?


Edit: Looks like Dr. D beat me to the punch!