Integrating f′(x)/f(x) — CAIE A-Level Mathematics

Introduction

A surprisingly large family of integrals — including $\int \tan x \, dx$ and $\int \dfrac{x}{x^2+1} \, dx$ — can all be resolved by a single, elegant observation: the numerator is (a constant multiple of) the derivative of the denominator. Recognising this structure instantly unlocks the result $\ln|f(x)| + c$ , with no substitution working required (though the logic behind it is a reverse chain rule). In 9709 examinations, these integrals appear both as standalone mark-scorers and embedded inside larger problems (integration by parts, differential equations, partial fractions). Mastering the pattern is essential.

Core Concept

Recall the chain rule for differentiation:

\frac{d}{dx}\bigl[\ln|f(x)|\bigr] = \frac{f'(x)}{f(x)}

Reversing this gives the fundamental result:

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

More generally, if the numerator is a constant multiple $k$ of $f'(x)$ :

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

The skill the examiner tests is recognition. Given an integrand, you must:

Identify $f(x)$ (the denominator).
Differentiate it to find $f'(x)$ .
Check whether the numerator is exactly $k \cdot f'(x)$ for some constant $k$ .
Write down the answer immediately.

If the numerator is almost $f'(x)$ but off by a constant factor, simply adjust $k$ to compensate — this is called adjusting the coefficient.

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Introduction

Core Concept

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