Indefinite Integration
Special Forms
Grade 12
Question:
<p>Evaluate \(\int \frac{\cos x + x \sin x}{x(x + \cos x)} dx\)</p>
<p>(A) \(\ln \frac{x + \cos x}{x} + C\)</p>
<p>(B) \(\ln \frac{x + \cos x}{x \sin x} + C\)</p>
<p>(C) \(\ln \frac{x \sin x}{x + \cos x} + C\)</p>
<p>(D) none of these</p>
Step-by-Step Solution
Key Concept: Recognize that this is of the form $\int \frac{f'(x)}{f(x)} dx = \ln|f(x)| + C$.
<p>Notice that the numerator $\cos x + x \sin x$ is the derivative of the denominator $x + \cos x$. Apply the logarithmic integration formula.</p>
Correct Answer: A