Question

The value of : \( \int \frac{x + 1}{x(1 + xe^x)} dx \).

• A. \( \log \left| \frac{1 + xe^x} {xe^x} \right| + C \)
• B. \( \log \left| \frac{xe^x}{1 + xe^x} \right| + C \)
• C. \( \log \left| xe^x(1 + xe^x) \right| + C \)
• D. \( \log \left| 1 + xe^x \right| + C \)

Hint:

When solving logarithmic integrals, identify substitutions that transform the integrand into simpler rational expressions.

Answer 1

The Correct Option is B

Consider the integral: \[ I = \int \frac{x + 1}{x(1 + xe^x)} \, dx. \] Step 1: Apply substitution. Let \( xe^x = t \), which implies that \( x + 1 = t \), and also \( dx = \frac{dt}{e^x} \). Step 2: Transform the integral. Substituting into the integral, we get: \[ I = \int \frac{dt}{t(1 + t)}. \] Step 3: Simplify the integrand. We can break the integrand into two simpler fractions: \[ I = \int \left( \frac{1}{t} - \frac{1}{1 + t} \right) \, dt. \] Step 4: Integrate. The integral of each term is straightforward: \[ I = \log |t| - \log |1 + t| + C. \] Step 5: Substitute back \( t = xe^x \). Substitute \( t = xe^x \) to return to the original variable: \[ I = \log \left| \frac{xe^x}{1 + xe^x} \right| + C. \] Final Answer: \[ \boxed{\log \left| \frac{xe^x}{1 + xe^x} \right| + C} \]