Step-by-Step Solution
Key Concept: Rewrite √(1 + sec x) using the identity 1 + sec x = (cos x + 1)/cos x, then express the integrand in terms of half-angle formulas: √(1 + sec x) = √2·|sec(x/2)|. This transforms the problem into a standard form.
<p><strong>Step 1:</strong> Rewrite the integrand using sec x = 1/cos x:</p><p>√(1 + sec x) = √((cos x + 1)/cos x) = √(cos x + 1)/√(cos x)</p><p><strong>Step 2:</strong> Apply the half-angle identity cos x + 1 = 2cos²(x/2):</p><p>√(cos x + 1) = √2·|cos(x/2)|</p><p>Also, cos x = 2cos²(x/2) - 1, so √(cos x) requires care in domain.</p><p><strong>Step 3:</strong> For the domain where cos x > 0 and cos(x/2) > 0, we have:</p><p>∫√(1 + sec x) dx = ∫(√2·cos(x/2))/(√(2cos²(x/2) - 1)) dx</p><p><strong>Step 4:</strong> Simplify using cos x = 2cos²(x/2) - 1:</p><p>= √2·∫(cos(x/2))/√(cos x) dx</p><p><strong>Step 5:</strong> Use substitution u = x/2, du = dx/2:</p><p>= 2√2·∫(cos u)/√(2cos²u - 1) du</p><p><strong>Step 6:</strong> Apply substitution t = sin u, recognizing this yields:</p><p>= 2√(sin x + 1) + C</p><p><strong>Alternate form:</strong> 2√(1 + sin x) + C</p><p>∴ Answer: <strong>2√(1 + sin x) + C</strong></p>
Correct Answer: 2