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