What is the antiderivative of e^2x/1+e^2x?

Answer 1

Please respect the order of operations -- exponentiation is done before multiplication, and division is done BEFORE addition. I suspect what you meant to write was:

e^(2x)/(1+e^(2x))

But what you actually wrote was:

e² * x/1 + e² * x, which is 2e²x, and has the obvious antiderivative e²x².

Anyway, proceeding with the function you meant to write:

∫e^(2x)/(1+e^(2x)) dx

First make the substitution u=1+e^(2x), du=2e^(2x) dx. So this integral becomes:

1/2 ∫1/u du

Which integrates easily as:

ln |u|/2

Finally:

ln |1+e^(2x)|/2

And we are done.

Answer 2

⌠
| (e^(2 x)+1)/2
⌡

oh... and pascal, the numbers don't have to be absolute values...

Answer 3

that's why you should do your homework!