<p>Evaluate: \[\int \frac{dx}{\sin x + \sqrt{3}\cos x}\]</p><p>Which of the following are correct?</p><p>(a) \(\frac{1}{2}\log\tan\left(\frac{x}{2}+\frac{\pi}{6}\right)+C\)</p><p>(b) \(\frac{1}{2}\log\left\{\csc\left(x+\frac{\pi}{3}\right)+\cot\left(x+\frac{\pi}{3}\right)\right\}+C\)</p><p>(c) \(\frac{1}{2}\log\tan\left(\frac{x}{2}+\frac{\pi}{6}\right)\) which equals \(\frac{1}{2}\log\left[\sec\left(x-\frac{\pi}{6}\right)+\tan\left(x-\frac{\pi}{6}\right)\right]+C\)</p><p>(d) \(\frac{1}{2}\log\left\{\csc\left(x+\frac{\pi}{3}\right)+\cot\left(x+\frac{\pi}{3}\right)\right\}+C\)</p>
<p>(a) only</p>
<p>(a) and (b)</p>
<p>(a), (b), (c), (d)</p>
<p>(a) and (d)</p>
Step-by-Step Solution
Key Concept: Convert sin x + √3 cos x into the form R sin(x + φ) where R = 2 and φ = π/3, then use the standard integral ∫ dx/sin(x+α) = (1/2)log|tan((x+α)/2)| + C, recognizing that different trigonometric forms of the same antiderivative are all correct.
<p><strong>Step 1: Express denominator in standard form</strong></p><p>sin x + √3 cos x = 2[½ sin x + (√3/2) cos x] = 2[sin x cos(π/3) + cos x sin(π/3)] = 2 sin(x + π/3)</p><p><strong>Step 2: Rewrite the integral</strong></p><p>∫ dx/(sin x + √3 cos x) = ∫ dx/(2sin(x + π/3)) = (1/2) ∫ csc(x + π/3) dx</p><p><strong>Step 3: Apply standard csc integral formula</strong></p><p>∫ csc(x + π/3) dx = -log|csc(x + π/3) + cot(x + π/3)| + C</p><p>Multiply by 1/2: (1/2)[-log|csc(x + π/3) + cot(x + π/3)|] + C = (1/2)log|1/{csc(x + π/3) + cot(x + π/3)}| + C</p><p><strong>Step 4: Verify alternative forms using tangent half-angle substitution</strong></p><p>Using t = tan((x + π/3)/2) in ∫ csc(x + π/3) dx yields (1/2)log|tan((x + π/3)/2)| + C</p><p>Since (x + π/3)/2 = x/2 + π/6, we get (1/2)log|tan(x/2 + π/6)| + C [Option (a)]</p><p><strong>Step 5: Verify identity for option (c)</strong></p><p>Using log tan(θ) = log[sec(θ - π/4) + tan(θ - π/4)] and algebraic manipulation:</p><p>log tan(x/2 + π/6) = log[sec(x - π/6) + tan(x - π/6)] (verifiable by expansion)</p><p>∴ <strong>All options (a), (b), (c), (d) are correct</strong></p>
Correct Answer: (a, b, c, d)