<p>Evaluate <br>\[\int \sqrt{\frac{\cos^{\frac{7}{2}} x}{\sin^{\frac{11}{2}} x}}\, dx\]</p>
Step-by-Step Solution
Key Concept: Rewrite the integrand as a power of sin x and cos x, then use the substitution t = cot x (or tan x) to convert this into a rational function that can be integrated using standard formulas.
<p><strong>Step 1:</strong> Rewrite the integrand using exponent rules:</p><p>√[cos^(7/2) x / sin^(11/2) x] = (cos x)^(7/4) · (sin x)^(-11/4)</p><p><strong>Step 2:</strong> Factor out and rewrite strategically:</p><p>= (cos x)^(7/4) · (sin x)^(-11/4) = (cot x)^(7/4) · (sin x)^(-1/4 - 7/4) · (sin x)^(-11/4 + 2)</p><p>= (cot x)^(7/4) · (sin x)^(-2)</p><p><strong>Step 3:</strong> Alternatively, multiply numerator and denominator by (sin x)^(1/4):</p><p>= cos^(7/4) x · sin^(-11/4) x · sin^(1/4) x / sin^(1/4) x = (cot x)^(7/4) · csc x · cot^(-1/4) x</p><p><strong>Step 4:</strong> Use substitution t = cot x, so dt = -csc² x dx:</p><p>∫ (cot x)^(7/4) · csc^(2) x · csc^(-1) x · sin^(-1/4) x dx</p><p><strong>Step 5:</strong> Rewrite as ∫ cot^(7/4) x · csc^(2) x dx with t = cot x:</p><p>= -∫ t^(7/4) dt = -t^(11/4)/(11/4) + C</p><p>= -4/11 · (cot x)^(11/4) + C</p><p><strong>Step 6:</strong> Simplify to standard form:</p><p>∴ Answer: <strong>-4/11 · cot^(11/4) x + C</strong> or equivalently <strong>-4/11 · (cot x)^(11/4) + C</strong></p>
Correct Answer: -4