Indefinite Integration
Substitution Method for Rational Expressions
Grade 12
Question:
<p>Evaluate $\int \frac{dx}{(x+3)^{15/16}(x-4)^{17/16}}$</p>
<p>(a) $\left(\frac{x+3}{x-4}\right)^{1/16} + C$</p>
<p>(b) $-\frac{1}{7}\left(\frac{x+3}{x-4}\right)^{1/16} + C$</p>
<p>(c) $\frac{1}{5}\left(\frac{x-4}{x+3}\right)^{1/16} + C$</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Use substitution $t = \frac{x+3}{x-4}$ to transform the integrand into a simpler power function form.
<p><strong>Solution:</strong> Let $I = \int \frac{dx}{(x+3)^{15/16}(x-4)^{17/16}}$</p><p>Rewrite as: $I = \int \frac{dx}{(x-4)^2 \left(\frac{x+3}{x-4}\right)^{15/16}}$</p><p>Let $\frac{x+3}{x-4} = t$</p><p>Then $\frac{d}{dx}\left(\frac{x+3}{x-4}\right) = \frac{(x-4) - (x+3)}{(x-4)^2} = \frac{-7}{(x-4)^2}$</p><p>So $(x-4)^{-2}dx = -\frac{1}{7}dt$</p><p>Therefore: $I = \int t^{-15/16} \cdot \left(-\frac{1}{7}\right)dt = -\frac{1}{7} \cdot \frac{t^{1/16}}{1/16} = -\frac{1}{7}\left(\frac{x+3}{x-4}\right)^{1/16} + C$</p><p>∴ Answer is (b).</p>
Correct Answer: B