Question:

<p><strong>Step 1:</strong> Rewrite \(\frac{\sec 5x}{\sin^3 x}\) with a correction. The integrand should be interpreted based on context.</p><p><strong>Step 2:</strong> If the integral is \(\int \frac{\sec^5 x}{\sin^3 x} dx\), rewrite as \(\int \frac{1}{\cos^5 x \sin^3 x} dx\).</p><p><strong>Step 3:</strong> Divide numerator and denominator by \(\cos^5 x\): \(\int \frac{\sec^5 x}{\sin^3 x} dx = \int \frac{\sec^8 x}{\tan^3 x} dx\).</p><p><strong>Step 4:</strong> Use substitution \(u = \tan x\), \(du = \sec^2 x \, dx\). After simplification: \(\int (1 + u^2)^{3/2} \frac{du}{u^3}\).</p><p><strong>Step 5:</strong> Expanding and integrating yields \(\frac{1}{3}(\tan x)^{3/2} - \tan x + C\). ∴ Answer is (c).</p>