Question

\[\int\frac{e^x}{\left( 1 + e^x \right)\left( 2 + e^x \right)} dx\]

Answer 1

Let ex = t therefore ex dx = dt
`int    e^x/[( 1 + e^x)( 2 + e^x )]dx  = int dt/[( 1 + t)( 2 + t)]`

 = `int dt/( 1 + t) - int dt/( 2 + t)`

= log| 1 + t | - log| 2 + t | + c

= log `|( 1 + t )/( 2 + t )| + c`

= log `|( 1 + e^x )/( 2 + e^x )|`+ c