Integrating f'(x)/f(x)

• Reverse Chain Rule Logic: This method is fundamentally the reverse of the chain rule applied to $\frac{d}{dx}(\ln|u|)$, which results in $\frac{1}{u} \cdot \frac{du}{dx}$.

• Logarithmic Derivative Relationship: Since differentiating $\ln|f(x)|$ yields $\frac{f'(x)}{f(x)}$ by the chain rule, it follows that the integral of the latter must return the former.

• Linearity and Constants: Integration is a linear operator, allowing us to factor out constants that may prevent the numerator from being an exact derivative initially.

• General Formula: The relationship is mathematically codified as:

$\int \frac{f'(x)}{f(x)} \, dx = \ln|f(x)| + C$

• Step 1: Structural Identification: Observe the fraction and identify the expression in the denominator as $f(x)$, then mentally calculate its derivative $f'(x)$.

• Step 2: Comparison and Adjustment: Compare the actual numerator to your calculated $f'(x)$; if they differ only by a constant factor $k$, you can manipulate the integral.

• Step 3: Compensate for Constants: Insert the required constant $k$ inside the integral to match $f'(x)$ and place its reciprocal $1/k$ outside the integral to maintain equality.

• Step 4: Final Integration: Apply the logarithmic rule to the adjusted integral and multiply the final result by the external compensation factor.

Example Strategy

• Scenario: Integrating $\frac{x}{x^2+5}$.
• Logic: Denominator derivative is $2x$. The numerator has $x$, so we need a factor of $2$.
• Execution: Write as $\frac{1}{2} \int \frac{2x}{x^2+5} \, dx$, resulting in $\frac{1}{2} \ln|x^2+5| + C$.

• Comparison of Fraction Types: It is vital to distinguish between fractions requiring logarithmic integration and those requiring the power rule or partial fractions.

• Recognition vs. Substitution: While this method is a 'shortcut' for the reverse chain rule, it is logically identical to using the substitution $u = f(x)$, which would eventually yield $\int \frac{1}{u} du$.

• The 'Bottom-First' Rule: Always begin by differentiating the denominator of any fraction you encounter in an integration question to see if it links to the numerator.

• Coefficient Verification: Examiners often provide a numerator that is missing a simple integer or fractional factor; always double-check the constant before writing the final $\ln$ term.

• Verify by Differentiation: A powerful way to check your answer is to differentiate your result; you should immediately arrive back at the original integrand.

• Modulus Marks: In many marking schemes, the modulus bars $|...|$ around the function in the $\ln$ term are required for full marks; do not omit them.

• Misapplying to Higher Powers: A common error is attempting to use the $\ln$ result for integrals like $\int \frac{1}{x^2} dx$; this requires the power rule ($x^{-2}$), not logarithms.

• Incorrect Correction Factors: Students often multiply by the constant instead of its reciprocal (e.g., using $3$ instead of $1/3$ outside the integral) when adjusting the numerator.

• Neglecting the '+C': Because these results often look like standard formulas, students frequently forget the constant of integration, which is required for indefinite integrals.

• Ignoring Non-Linear Inner Functions: Assuming $\int \frac{1}{ax+b} dx$ is just $\ln|ax+b|$ without dividing by the coefficient $a$ is a frequent source of lost marks.

• Trigonometric Integrals: Many standard results, such as the integral of $\tan(x)$, are derived using this method by rewriting the function as $\frac{\sin(x)}{\cos(x)}$.

• Partial Fractions Connection: When a complex rational function is split into partial fractions, the resulting terms are often in the form $\frac{A}{x+B}$, which are integrated using this $f'(x)/f(x)$ principle.

• Differential Equations: This pattern frequently appears when solving first-order separable differential equations where variables are grouped into quotients.