Question

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

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

Hint:

When solving integrals of rational functions, recognize standard forms for the integrals. The integral \( \int \frac{dx}{1 + e^x} \) directly leads to the logarithmic form \( \log|1 + e^x| + C \).

Answer 1

The Correct Option is B

We are tasked with solving the integral: \[ \int \frac{dx}{1 + e^x} \] This integral is of a standard form. We recognize that the integral of \( \frac{1}{1 + e^x} \) is directly related to the natural logarithm function. 
Specifically: \[ \int \frac{dx}{1 + e^x} = \log|1 + e^x| + C \] where \( C \) is the constant of integration. Thus, the correct answer is \( \log|1 + e^x| + C \).