`(i) int(e^(x))/(1+e^(x))dx" "(ii) int (e^(x)) /((1+e^(x))^(4))dx`

Answer

Let's solve the integrals step by step.

### (i) Integral of \( \frac{e^x}{1 + e^x} \, dx \)

1. **Substitution**: Let \( t = 1 + e^x \). Then, differentiate both sides:
\[ dt = e^x \, dx \quad \Rightarrow \quad dx = \frac{dt}{e^x} \]
Since \( e^x = t - 1 \), we can substitute:
\[ dx = \frac{dt}{t - 1} \]

2. **Rewrite the integral**: Substitute \( t \) into the integral:
\[ \int \frac{e^x}{1 + e^x} \, dx = \int \frac{e^x}{t} \cdot \frac{dt}{e^x} = \int \frac{1}{t} \, dt \]

3. **Integrate**: The integral of \( \frac{1}{t} \) is:
\[ \int \frac{1}{t} \, dt = \ln |t| + C \]

4. **Back-substitute**: Replace \( t \) back with \( 1 + e^x \):
\[ \ln |1 + e^x| + C \]

Thus, the final answer for the first integral is:
\[ \int \frac{e^x}{1 + e^x} \, dx = \ln(1 + e^x) + C \]