Indefinite Integration
Integration of Fractional Powers
Grade 12
Question:
<p>The integral <span class="math">\(\int \frac{dx}{(x + 4)^{8/7}(x - 3)^{6/7}}\)</span> is equal to (where C is a constant of integration)</p>
<p>(a) <span class="math">-\frac{1}{(x-3)^{1/7}}{(x+4)^{1/7}}\)</span> + C</p>
<p>(b) <span class="math">\frac{1}{2}\left(\frac{x-3}{x+4}\right)^{1/7}\)</span> + C</p>
<p>(c) <span class="math">\frac{1}{2}\left(\frac{x-3}{x+4}\right)^{-13/7}\)</span> + C</p>
<p>(d) <span class="math">-\frac{13}{7}\left(\frac{x-3}{x+4}\right)^{1/7}\)</span> + C</p>
Step-by-Step Solution
Key Concept: Recognize the structure of the integrand involving fractional powers and use appropriate algebraic manipulation or substitution to simplify the expression.
<p><strong>Solution:</strong> The given integral,</p><p><span class="math">$I = \int \frac{dx}{(x + 4)^{8/7}(x - 3)^{6/7}}$</span></p><p>This can be rewritten and solved using substitution methods for fractional powers, yielding the result <span class="math">$\frac{1}{2}\left(\frac{x-3}{x+4}\right)^{-13/7}$</span> + C</p>
Correct Answer: c