<p>The value of \(\int_0^1 (1 + e^{-x^2}) dx\) is</p>
Step-by-Step Solution
Key Concept: This integral cannot be expressed in closed form using elementary functions because e^(-x²) lacks an elementary antiderivative. The solution requires recognizing that the answer involves the error function erf(x), defined as erf(x) = (2/√π)∫₀ˣ e^(-t²)dt.
<p><strong>Step 1:</strong> Recognize that ∫₀¹ (1 + e^(-x²))dx = ∫₀¹ 1·dx + ∫₀¹ e^(-x²)dx</p><p><strong>Step 2:</strong> The first integral: ∫₀¹ 1·dx = [x]₀¹ = 1</p><p><strong>Step 3:</strong> The second integral ∫₀¹ e^(-x²)dx cannot be expressed using elementary functions. It relates to the error function: ∫₀¹ e^(-x²)dx = (√π/2)·erf(1)</p><p><strong>Step 4:</strong> Therefore: ∫₀¹ (1 + e^(-x²))dx = 1 + (√π/2)·erf(1), where erf(1) ≈ 0.8427</p><p><strong>Step 5:</strong> This gives approximately 1 + (√π/2)(0.8427) ≈ 1 + 0.747 ≈ 1.747</p><p>∴ Answer: D (typically expressed as 1 + (√π/2)erf(1) or numerically ≈ 1.747)</p>
Correct Answer: D