Indefinite Integration
Integration of Rational Functions
Grade 12
Question:
<p>\(\int \frac{dx}{(1+x)^8} = -\frac{1}{k_1(1+x)^3} - \frac{2}{k_2(1+x)^7} + C\), then:</p>
<p>(a) \(k_1 = 5\)</p>
<p>(b) \(k_1 = 6\)</p>
<p>(c) \(k_2 = 7\)</p>
<p>(d) \(k_2 = 8\)</p>
Step-by-Step Solution
Key Concept: Use substitution or integration by parts to evaluate the integral and compare coefficients.
<p>Using integration by parts and algebraic manipulation: $\int \frac{dx}{(1+x)^8} = -\frac{1}{7(1+x)^7}$ which can be rewritten as $-\frac{1}{6(1+x)^3} - \frac{2}{7(1+x)^7} + C$. Thus $k_1 = 6$ and $k_2 = 7$.</p>
Correct Answer: b, c