Indefinite Integration
Integration of Series
Grade 12
Question:
<p>If \(f(x) = \cos x - \cos 2x + \cos 3x - \ldots \infty\), then \(\int f(x) dx\) is equal to</p>
<p>(a) \(\tan \frac{x}{2} + C\)</p>
<p>(b) \(x - \tan \frac{x}{2} + C\)</p>
<p>(c) \(x - \tan \frac{x}{2} + C\)</p>
<p>(d) \(\tan \frac{x}{2} + C\)</p>
Step-by-Step Solution
Key Concept: Sum the alternating infinite series of cosines, then integrate the resulting expression.
<p>The infinite series $f(x) = \cos x - \cos 2x + \cos 3x - \ldots$ can be summed using the formula for alternating cosine series, which gives $f(x) = \frac{\sin x}{1 + 2\sin x}$ or equivalent form. Integration yields $\int f(x) dx = x - \tan \frac{x}{2} + C$.</p>
Correct Answer: B