Integrating Rational Functions of the Form f'(x)/f(x)

Task goals

• Foundation: Integrate simple rational functions where the numerator is exactly the derivative of the denominator, e.g. recognising forms like \int \frac{f'(x)}{f(x)}\,dx = \ln|f(x)| + c.
• Proficient: Manipulate the numerator with a constant multiplier or simple algebraic adjustment before applying the ln|f(x)| rule, including definite integrals evaluated between two limits.
• Excellence: Solve multi-step problems combining this technique with other integration methods (e.g. splitting a fraction, substitution) or applying it within a real-world rate-of-change or area context requiring justification of domain restrictions.