Definite Integration
Finding Limits and Integrating
Grade 12
Question:
<p>If <span class="math">\theta_1</span> and <span class="math">\theta_2</span> be respectively the smallest and the largest values of <span class="math">\theta</span> in <span class="math">(0, 2\pi) - \{\pi/2\}</span> which satisfy the equation <span class="math">2\cot\theta - \frac{5}{\sin\theta} + 4 = 0</span>, then <span class="math">\int_{\theta_1}^{\theta_2} \cos^2 3\theta d\theta</span> equals</p>
<p>(a) <span class="math">\frac{\pi}{3}</span></p>
<p>(b) <span class="math">\frac{1}{6}</span></p>
<p>(c) <span class="math">\frac{\pi}{3}</span></p>
<p>(d) <span class="math">\frac{2\pi}{3}</span></p>
Step-by-Step Solution
Key Concept: Solve the trigonometric equation for the limits, then use the double angle formula to simplify the integrand.
<p>First solve <span class="math">2\cot\theta - \frac{5}{\sin\theta} + 4 = 0</span> to find <span class="math">\theta_1</span> and <span class="math">\theta_2</span>. Then evaluate <span class="math">\int_{\theta_1}^{\theta_2} \cos^2 3\theta d\theta</span> using the identity <span class="math">\cos^2 3\theta = \frac{1 + \cos 6\theta}{2}</span>.</p>
Correct Answer: C