Question

How do you determine if the improper integral converges or diverges `int (e^x)/(e^2x + 1) dx` from 0 to infinity?

Answer 1

The integral converges, and:

`int_0^oo \ (e^x)/(e^(2x)+1) \ dx = pi/4`

Explanation:

Assuming a correction, let us first find:

`I = int \ (e^x)/(e^(2x)+1) \ dx`

The utilising a substitution if `u=e^x => (du)/dx=e^x` yields:

`I = int \ (1)/(u^2+1) \ du`

Which is a standard integral, and so integrating:

`I = arctan (u) + C`

And then reversing the earlier substitution:

`I = arctan (e^x) + C`

If we look at the graph of the function, it would appear as if the bounded area is finite:

graph{(e^x)/(e^(2x)+1) [-10, 10, -2, 2]}

So let us test this prediction analytically. Consider the improper definite integral:

`J = int_0^oo \ (e^x)/(e^(2x)+1) \ dx`

`\ \ = lim_(n rarr oo) [arctan (e^x)]_0^n`

`\ \ = lim_(n rarr oo) (arctan (e^n)) -arctan (e^0)`

Now, as `x rarr oo => e^x rarr oo`, thus:

`J = lim_(n rarr oo) (arctan n) - arctan (1)`

`\ \ = pi/2-pi/4`

`\ \ = pi/4`