Question

If \( \int_0^1 \frac{e^x}{1+x} \, dx = \alpha \), then \( \int_0^1 \frac{e^x}{(1+x)^2} \, dx \) is equal to:

Hint:

For integrals involving rational functions with exponential terms, integration by parts or substitution can help simplify the problem.

Answer 1

We are given: \[ \int_0^1 \frac{e^x}{1+x} \, dx = \alpha \] We need to compute the integral: \[ \int_0^1 \frac{e^x}{(1+x)^2} \, dx \] Using integration by parts or substitution, we obtain: \[ \int_0^1 \frac{e^x}{(1+x)^2} \, dx = \alpha - 1 - \frac{e}{2} \] Thus, the correct answer is (C) \( \alpha - 1 - \frac{e}{2} \).