<p>Evaluate: \ 11\int \frac{\sec^2 x}{(\sec x + \tan x)^n} \, dx, \quad (n > 1)</p>
Step-by-Step Solution
Key Concept: Recognize that the derivative of (sec x + tan x) is sec x(sec x + tan x), allowing a u-substitution where u = sec x + tan x to transform the integral into a standard power form.
<p><strong>Step 1:</strong> Let u = sec x + tan x</p><p><strong>Step 2:</strong> Find du: du = (sec x tan x + sec²x) dx = sec x(tan x + sec x) dx = sec x · u · dx</p><p>Therefore: sec²x dx = (sec x · sec x) dx. We need to express sec²x dx in terms of du.</p><p><strong>Step 3:</strong> Note that du = sec x(sec x + tan x) dx, so: sec x dx = du/(sec x + tan x) = du/u</p><p>Thus: sec²x dx = sec x · sec x dx = sec x · du/u</p><p><strong>Step 4:</strong> Alternatively, recognize: d(sec x + tan x) = sec x(sec x + tan x) dx</p><p>So the integral becomes: ∫ sec²x/(sec x + tan x)^n dx = ∫ (1/u^n) · (du/u) is incorrect.</p><p><strong>Correct approach:</strong> Let u = sec x + tan x, then du = sec x(sec x + tan x) dx</p><p>∫ sec²x/(sec x + tan x)^n dx = ∫ sec x · sec x/(sec x + tan x)^n dx</p><p>Since du = sec x · u · dx, we have sec x dx = du/u</p><p>Therefore: ∫ sec x · sec x dx/u^n = ∫ du/u^(n+1)</p><p><strong>Step 5:</strong> Integrate: ∫ u^(-(n+1)) du = u^(-n)/(-n) + C = -1/(n·u^n) + C</p><p><strong>Step 6:</strong> Substitute back: u = sec x + tan x</p><p>∴ Answer: <strong>-1/[n(sec x + tan x)^n] + C</strong> or equivalently <strong>-1/[n·u^n] + C</strong></p>
Correct Answer: -1