Integrating 1/x and Related Reciprocal Functions

Task goals

• Foundation: Recognise that $\int \frac{1}{x}\,dx = \ln|x| + C$ and apply it to simple constant-multiple integrals.
• Proficient: Integrate expressions of the form $\frac{1}{ax+b}$ using the adjusted log rule, and evaluate definite integrals involving $\ln$, attending to domain restrictions.
• Excellence: Integrate rational functions that require algebraic manipulation (e.g. splitting a fraction or simple substitution) before the $1/x$ rule applies, and interpret results such as area under a reciprocal curve.