Your equations seem to have a few errors. I think what you meant is this:

\(\displaystyle

\int _ 0 ^ X \frac {2 \lambda e ^{- \lambda x}} {(1 + e ^{- \lambda x} ) ^2 } dx

\)

Is that right? You are changing the variable from **x** to *u* using:

\(\displaystyle

u = \frac 1 {1 + e^ {-\lambda x}}

\)

From this you get:

\(\displaystyle

x = \frac {-1} {\lambda} ln( \frac {1-u} u)\), and \(\displaystyle dx = \frac {1} {\lambda u (1-u)} du \)

When you do this you have to change the limits of integration to conform to this new variable. So where initially you had a lower limit of *x* = 0, you now have \(\displaystyle u = \frac 1 {1 + e^0} = \frac 1 2 \). Likewise, the upper end becomes \(\displaystyle u = \frac 1 {1 + e ^ {- \lambda X}}\). And the new integral is:

\(\displaystyle

2\int _ {\frac 1 2 } ^ \frac 1 {1 + e ^ {- \lambda X}} du

\)