Indefinite Integration
Trigonometric integrals
Grade 12
Question:
<p>Given<br>\[ I = \int \frac{\sin\left(\dfrac{5x}{2}\right)}{\sin\left(\dfrac{x}{2}\right)}\, dx \]<br>Evaluate \(I\).</p>
<p>\(2\sin x + x + \sin 2x + C\)</p>
<p>\(2\sin x + x - \sin 2x + C\)</p>
<p>\(2\sin x + x + \sin 2x + C\)</p>
<p>\(\sin x + x + \sin 2x + C\)</p>
Step-by-Step Solution
Key Concept: Use the sum-to-product identity or expand the numerator using sin(5x/2) = sin(2x + x/2) repeatedly, recognizing that sin(nθ)/sin(θ) can be expressed as a Chebyshev polynomial or by systematic expansion using sin(A+B). The key is to convert the ratio into a polynomial in cos(x/2) plus a remainder term.
<p><strong>Step 1:</strong> Recognize that sin(5x/2)/sin(x/2) can be simplified using the identity for sin(nθ)/sin(θ). Use the fact that:</p><p>sin(5θ)/sin(θ) = 4cos(2θ) + 2 (derived via Chebyshev or product expansion)</p><p><strong>Step 2:</strong> Substitute θ = x/2, so 2θ = x:</p><p>sin(5x/2)/sin(x/2) = 4cos(x) + 2</p><p><strong>Step 3:</strong> Integrate term by term:</p><p>I = ∫(4cos(x) + 2)dx = 4sin(x) + 2x + C</p><p>∴ Answer: <strong>I = 4sin(x) + 2x + C</strong></p>
Correct Answer: A