Definite Integration
Complex Trigonometric Integrands
Grade 12
Question:
<p>The integral <span class="math">\int_{\pi/6}^{\pi/4} \frac{dx}{\sin^2 x(\tan^5 x + \cot^5 x)}</span> is equal to</p>
<p>(a) <span class="math">\frac{1}{5}\left(\frac{\pi}{4} - \tan^{-1}\frac{1}{3\sqrt{3}}\right)</span></p>
<p>(b) <span class="math">\frac{1}{20}\tan^{-1}\frac{1}{9\sqrt{3}}</span></p>
<p>(c) <span class="math">\frac{\pi}{8} - \frac{1}{10}\tan^{-1}\frac{1}{9\sqrt{3}}</span></p>
<p>(d) Other</p>
Step-by-Step Solution
Key Concept: Express sums of trigonometric powers in terms of simpler functions and use appropriate substitution.
<p>Rewrite: <span class="math">\tan^5 x + \cot^5 x = \frac{\sin^5 x + \cos^5 x}{\sin^5 x \cos^5 x}</span></p><p>The integrand becomes: <span class="math">\frac{\sin^5 x \cos^5 x}{\sin^2 x(\sin^5 x + \cos^5 x)} = \frac{\sin^3 x \cos^5 x}{\sin^5 x + \cos^5 x}</span></p><p>Use substitution and trigonometric identities to evaluate.</p>
Correct Answer: C