Indefinite Integration
Integration by substitution/trigonometric integrals
Grade 12
Question:
<p>Evaluate: \[\int \frac{\sin x}{\sin\left(x - \dfrac{\pi}{4}\right)}\,dx\]</p><p>Which of the following are correct forms of the answer?</p>
<p>(a) \(x + \log\left|\sin\left(x-\dfrac{\pi}{4}\right)\right| + C\)</p>
<p>(b) \(x - \log\left|\sin\left(x-\dfrac{\pi}{4}\right)\right| + C\)</p>
<p>(c) \(x + \cos\left(x-\dfrac{\pi}{4}\right) + C\)</p>
<p>(d) \(x - \cos\left(x-\dfrac{\pi}{4}\right) + C\)</p>
Step-by-Step Solution
Key Concept: Express sin(x - π/4) using angle subtraction, then manipulate the numerator to match the denominator's structure. This allows splitting into separate integrable parts using the substitution strategy.
<p><strong>Step 1:</strong> Use angle subtraction formula: sin(x - π/4) = sin x cos(π/4) - cos x sin(π/4) = (sin x - cos x)/√2</p><p><strong>Step 2:</strong> Rewrite the integral:<br>∫ sin x / [(sin x - cos x)/√2] dx = √2 ∫ sin x / (sin x - cos x) dx</p><p><strong>Step 3:</strong> Express numerator as: sin x = A(sin x - cos x) + B(-cos x - sin x), which gives sin x = A(sin x - cos x) + B(d/dx)(sin x - cos x)<br>Comparing: A = 1/2 and B = 1/2</p><p><strong>Step 4:</strong> Split the integral:<br>√2 ∫ [1/2 · (sin x - cos x) + 1/2 · (sin x + cos x)] / (sin x - cos x) dx<br>= (√2/2) ∫ dx + (√2/2) ∫ (sin x + cos x)/(sin x - cos x) dx<br>= (√2/2)x + (√2/2) ∫ d(sin x - cos x)/(sin x - cos x)<br>= (√2/2)x + (√2/2) ln|sin x - cos x| + C<p><strong>Step 5:</strong> Alternative forms:<br>• (√2/2)[x + ln|sin x - cos x|] + C<br>• (1/√2)[x + ln|sin x - cos x|] + C<br>• (√2/2)x + (√2/2) ln|sin x - cos x| + C</p><p>∴ Answer: (b, d)</p>
Correct Answer: (b, d)