<p>Given \( I = \int_{\pi/6}^{\pi/3} \sec^{2/3} x \cdot \csc^{4/3} x \, dx \), find the value of \(I\).</p>
Step-by-Step Solution
Key Concept: Convert the integrand to a single trigonometric function using tan x substitution: sec^{2/3}x · csc^{4/3}x = tan^{4/3}x · sec^{2}x, then substitute u = tan x to get a power function integral.
<p><strong>Step 1:</strong> Rewrite the integrand using sec²x = 1 + tan²x identity.</p><p>sec^{2/3}x · csc^{4/3}x = sec^{2/3}x · (sin x)^{-4/3} · (cos x)^{4/3} · (cos x)^{-4/3}</p><p>= (cos x)^{-2/3} · (sin x)^{-4/3} = (tan x)^{-4/3} · (cos x)^{-2/3} · (cos x)^{4/3}</p><p>Simplifying: sec^{2/3}x · csc^{4/3}x = sec²x · tan^{4/3}x</p><p><strong>Step 2:</strong> Use substitution u = tan x, so du = sec²x dx.</p><p>When x = π/6: u = tan(π/6) = 1/√3 = 3^{-1/2}</p><p>When x = π/3: u = tan(π/3) = √3 = 3^{1/2}</p><p><strong>Step 3:</strong> Transform the integral:</p><p>I = ∫_{3^{-1/2}}^{3^{1/2}} u^{4/3} du</p><p><strong>Step 4:</strong> Integrate using power rule:</p><p>I = [u^{4/3 + 1}/(4/3 + 1)]_{3^{-1/2}}^{3^{1/2}} = [u^{7/3}/(7/3)]_{3^{-1/2}}^{3^{1/2}}</p><p>= (3/7)[(3^{1/2})^{7/3} - (3^{-1/2})^{7/3}]</p><p>= (3/7)[3^{7/6} - 3^{-7/6}]</p><p><strong>Step 5:</strong> Factor out 3^{1/6}:</p><p>= (3/7) · 3^{1/6}[3 - 3^{-4/3}] = (3/7) · 3^{1/6} · 3[3^{-1/3} - 3^{-5/6}]</p><p>∴ Answer: <strong>3[3^{1/6} - 3^{-5/6}]</strong></p>
Correct Answer: 3[3^{1/6} - 3^{-5/6}]