Indefinite Integration
Rational Functions
Grade 12
Question:
<p>If \(\int \frac{1 - x^7}{x(1 + x^7)} dx = P \log|x| + Q \log|x^7 + 1| + C\), then:</p>
<p>(a) \(2P - 7Q = 0\)</p>
<p>(b) \(2P + 7Q = 0\)</p>
<p>(c) \(7P + 2Q = 0\)</p>
<p>(d) \(7P - 2Q = 1\)</p>
Step-by-Step Solution
Key Concept: Partial fraction decomposition directly yields the coefficients; relate them through algebraic manipulation.
<p>Use partial fraction decomposition: $\frac{1-x^7}{x(1+x^7)} = \frac{A}{x} + \frac{B}{1+x^7}$. Multiply both sides by $x(1+x^7)$ and solve for $A$ and $B$. This gives $A = 1$ and the coefficient of $\log|1+x^7|$ determines $Q$. The relationship between $P$ and $Q$ follows from the decomposition.</p>
Correct Answer: C