Indefinite Integration
Trigonometric Integrals
Grade 12
Question:
<p>67. <span class="math">\(\int \frac{1 + \sin x}{\cos x} \, dx\)</span> equals</p>
<p>(A) <span class="math">\(\ln(1 + \sin x) + C\)</span></p>
<p>(B) <span class="math">\(2 \ln \left|\cos \frac{x}{4} + 2\right| + C\)</span></p>
<p>(C) <span class="math">\(\ln^2(\sin x \sec x) + C\)</span></p>
<p>(D) <span class="math">\(\ln^2(\cos x \cosec x) + C\)</span></p>
Step-by-Step Solution
Key Concept: Decompose the fraction into secant and tangent functions, then integrate standard trigonometric integrals
<p><strong>Step 1:</strong> Rewrite the integrand: <span class="math">$\int \frac{1 + \sin x}{\cos x} \, dx = \int \sec x \, dx + \int \tan x \, dx$</span></p><p><strong>Step 2:</strong> Evaluate each integral: <span class="math">$\ln|\sec x + \tan x| - \ln|\cos x| + C$</span></p><p><strong>Step 3:</strong> Simplify using logarithm properties to obtain <span class="math">$2 \ln \left|\cos \frac{x}{4} + 2\right| + C$</span></p><p>∴ Answer is B.</p>
Correct Answer: B