Indefinite Integration
Integration by Recognition
Grade 12
Question:
<p>If \(\int e^{\sec x}(\sec x \tan x + (\sec x \tan x + \sec^2 x))\,dx = e^{\sec x} f(x) + C\), then a possible choice of \(f(x)\) is</p>
<p>(a) \(x \sec x + \tan x + \frac{1}{2}\)</p>
<p>(b) \(\sec x + \tan x + \frac{1}{2}\)</p>
<p>(c) \(\sec x + x \tan x - \frac{1}{2}\)</p>
<p>(d) \(\sec x - \tan x - \frac{1}{2}\)</p>
Step-by-Step Solution
Key Concept: Recognize the integrand as a derivative of a product and match coefficients to find f(x).
<p>Using integration by parts and the product rule for differentiation in reverse, we identify \(f(x) = \sec x + \tan x + \frac{1}{2}\) satisfies the given condition.</p>
Correct Answer: B