Question

The value of \( \int \frac{1}{1 + e^{x}} \, dx \) is:

• A. $\log(1+e^{x}) + c$
• B. $-\log(1+e^{-x}) + c$
• C. $\log(1+e^{-x}) + c$
• D. $x - \log(1+e^{x}) + c$

Hint:

Adding and subtracting terms in the numerator is a powerful trick for rational integrals.

Answer 1

The Correct Option is D

Step 1: Concept
Multiply numerator and denominator by $e^{-x}$ or add/subtract $e^x$ in the numerator.
Step 2: Analysis
$\int \frac{1+e^x-e^x}{1+e^x} dx = \int (1 - \frac{e^x}{1+e^x}) dx = \int 1 dx - \int \frac{e^x}{1+e^x} dx$.
Step 3: Conclusion
$= x - \log(1+e^{x}) + c$.
Final Answer: (D)