Definite Integration
Fractional Exponents
Grade 12
Question:
<p>The integral <span class="math">\int_{\pi/6}^{\pi/3} \sec^{2/3} x \csc^{4/3} x dx</span> is equal to</p>
<p>(a) <span class="math">3^{5/6} - 3^{2/3}</span></p>
<p>(b) <span class="math">3^{7/6} - 3^{5/6}</span></p>
<p>(c) <span class="math">3^{5/3} - 3^{1/3}</span></p>
<p>(d) <span class="math">3^{4/3} - 3^{1/3}</span></p>
Step-by-Step Solution
Key Concept: Rewrite fractional powers of trigonometric functions in a form suitable for substitution.
<p>Rewrite the integrand: <span class="math">\sec^{2/3} x \csc^{4/3} x = \frac{\sec^2 x}{\sec^{4/3} x \csc^{4/3} x} = \frac{\sec^2 x}{(\tan x)^{4/3}}</span></p><p>Let <span class="math">u = \tan x</span>, then <span class="math">du = \sec^2 x dx</span></p><p>Evaluate the resulting integral with the transformed limits.</p>
Correct Answer: D