<p>Evaluate: \(\displaystyle\int_{-\pi/2}^{\pi/2} \frac{2}{1+e^x}\,dx\) = ______ (up to four decimal places).</p>
Step-by-Step Solution
Key Concept: Recognize that the integrand can be decomposed using the property f(x) + f(-x) = 2/(1+e^x) + 2/(1+e^(-x)) = 2, which simplifies the integral through symmetry arguments rather than direct antiderivative computation.
<p><strong>Step 1:</strong> Let I = ∫_{-π/2}^{π/2} 2/(1+e^x) dx</p><p><strong>Step 2:</strong> Use the property that for symmetric limits, consider f(-x):</p><p>f(-x) = 2/(1+e^(-x)) = 2e^x/(1+e^x)</p><p><strong>Step 3:</strong> Add f(x) + f(-x):</p><p>2/(1+e^x) + 2e^x/(1+e^x) = (2 + 2e^x)/(1+e^x) = 2(1+e^x)/(1+e^x) = 2</p><p><strong>Step 4:</strong> Therefore: ∫_{-π/2}^{π/2} [f(x) + f(-x)] dx = ∫_{-π/2}^{π/2} 2 dx</p><p><strong>Step 5:</strong> Since I = ∫_{-π/2}^{π/2} f(x) dx and by symmetry property:</p><p>2I = ∫_{-π/2}^{π/2} 2 dx = 2[x]_{-π/2}^{π/2} = 2(π/2 - (-π/2)) = 2π</p><p><strong>Step 6:</strong> Thus I = π ≈ 3.1416</p><p>∴ Answer: <strong>3.1416</strong></p>
Correct Answer: 3