Question:medium
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:
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:
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For integrals involving rational functions with exponential terms, integration by parts or substitution can help simplify the problem.
Updated On: Mar 8, 2026
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Solution and Explanation
Given that \( \int_0^1 \frac{e^x}{1+x} \, dx = \alpha \), we are asked to compute \( \int_0^1 \frac{e^x}{(1+x)^2} \, dx \). By applying integration by parts or substitution, this integral evaluates to \( \alpha - 1 - \frac{e}{2} \). Therefore, the correct answer is (C) \( \alpha - 1 - \frac{e}{2} \).
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