Expert: Paul Klarreich - 12/6/2006

Question
This ones got me stumped.  Thanks.

Integrate from inf to -inf     e^x
                             ------
                            1 + e^(2x)


Answer
Questioner:  Gary
Category:  Calculus
 
Subject:  Integration
Question:  This ones got me stumped.  Thanks.

Integrate from inf to -inf     e^x
                            ------
                           1 + e^(2x)

.................................
Hi, Gary,

To do the integration:

{   e^x dx
| ----------
} 1 + e^(2x)

Make the observation that  e^(2x) = (e^x)^2, and that D(e^x) = e^x.

That suggests the substitution:

u = e^x,  du = e^x dx

{    du
| -------- =
} 1 + u^2

That's just a standard form:

arctan u =
arctan (e^x) from  x = -inf to x = +inf.

Now when x -> -inf,  e^x --> 0, and  arctan(0) = 0

When x -> inf, e^x -> inf,
But when u -> inf,  arctan u --> pi/2

So the integral is pi/2