Indefinite Integration
Integration by Substitution
Grade 12
Question:
<p>Evaluate <span class="math">\(I = \int \frac{(\sin 2x)^{1/3}}{(\sin^{2/3} x + \cos^{2/3} x)^2} d(\tan^{1/3} x)\)</span></p>
<p>(a) <span class="math">\(\frac{1}{2\sqrt[3]{2}} \ln(1 + \tan^{1/3} x) + C\)</span></p>
<p>(b) <span class="math">\(\ln(1 + \tan^{2/3} x) + C\)</span></p>
<p>(c) <span class="math">\(2\sqrt[3]{2} \ln(1 + \tan^{1/3} x) + C\)</span></p>
<p>(d) <span class="math">\(\frac{1}{2\sqrt[3]{2}} \ln(1 + \tan^{2/3} x) + C\)</span></p>
Step-by-Step Solution
Key Concept: Recognize the differential form d(tan^{1/3}x) in the integrand and use it directly as a substitution hint.
<p>This is a substitution problem. The integrand structure suggests using the substitution relationship with the differential form provided.</p>
Correct Answer: A