Indefinite Integration
Integration of trigonometric functions
Grade 12
Question:
<p>\(\int \frac{dx}{\cos x + \sqrt{3}\sin x}\) equals</p>
<p>\(\frac{1}{2}\log\tan\left(\frac{x}{2}+\frac{\pi}{12}\right)+C\)</p>
<p>\(\frac{1}{2}\log\tan\left(\frac{x}{2}-\frac{\pi}{12}\right)+C\)</p>
<p>\(\log\tan\left(\frac{x}{2}+\frac{\pi}{12}\right)+C\)</p>
<p>\(\log\tan\left(\frac{x}{2}-\frac{\pi}{12}\right)+C\)</p>
Step-by-Step Solution
Key Concept: Rewrite the denominator as a single sinusoidal function using the form R·sin(x+φ) or R·cos(x+φ), then use the Weierstrass substitution t = tan(x/2) to convert to a rational function.
<p><strong>Step 1:</strong> Express denominator in standard form</p><p>cos x + √3 sin x = 2(1/2·cos x + √3/2·sin x) = 2(sin 30°·cos x + cos 30°·sin x) = 2 sin(x + 30°) = 2 sin(x + π/6)</p><p><strong>Step 2:</strong> Substitute into integral</p><p>∫ dx/(2 sin(x + π/6)) = (1/2)∫ csc(x + π/6) dx</p><p><strong>Step 3:</strong> Use standard csc integral formula</p><p>∫ csc u du = -ln|csc u + cot u| + C = ln|tan(u/2)| + C</p><p><strong>Step 4:</strong> Apply with u = x + π/6</p><p>= (1/2)[-ln|csc(x + π/6) + cot(x + π/6)|] + C</p><p>= (1/2) ln|tan(x/2 + π/12)| + C</p><p><strong>Alternative form:</strong> (-1/2)ln|csc(x + π/6) + cot(x + π/6)| + C</p><p>∴ Answer: A</p>
Correct Answer: A