Indefinite Integration
Rational Functions
Grade 12
Question:
<p>If \(I = \int \frac{dx}{(x-1)^4(x+2)^3} = k\sqrt[4]{x+2} + C\), then \(k\) is equal to:</p>
<p>(a) \(\frac{1}{3}\)</p>
<p>(b) \(\frac{2}{3}\)</p>
<p>(c) \(\frac{3}{4}\)</p>
<p>(d) \(\frac{4}{3}\)</p>
Step-by-Step Solution
Key Concept: Partial fractions combined with careful observation of the resulting form reveals the coefficient.
<p>Use partial fraction decomposition to break down the rational function. The key observation is recognizing the form that leads to a fourth root. Differentiate the answer to find $k$.</p>
Correct Answer: A