Integrate the functions`(e^x)/((1+e^x)(2+e^x)`

Explanation

To integrate the function \(\frac{e^x}{(1+e^x)(2+e^x)}\), we will use the substitution method and partial fraction decomposition. Let's go through the steps in detail:

### Step 1: Substitution
Let \( t = e^x \). Then, the differential \( dx \) can be expressed as:
\[ dx = \frac{dt}{t} \]
This means that \( e^x dx = dt \). Now we can rewrite the integral:
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