k f′(x) / f(x): the missing numerator trick

8 minA Level3.5.4

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can spot fractions where the top is (a multiple of) the derivative of the bottom, like $\dfrac{x}{x^2 + 1}$ , and integrate them straight to a log.

Key to the symbols

Symbol	Meaning
$f(x)$	any function of $x$ , here the bottom of the fraction
$f'(x)$	the derivative of $f(x)$
$k$	a constant number
$\lvert f(x) \rvert$	the size of $f(x)$ , ignoring any minus sign

Definition

The f-over-f rule

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

In words: if the top is the derivative of the bottom, the answer is ln of the bottom.

Why it's true: differentiate $\ln|f(x)|$ using the chain rule. The outside, $\ln$ , gives $\dfrac{1}{f(x)}$ . Then multiply by the inside's derivative, $f'(x)$ :

$\frac{d}{dx}\ln|f(x)| = \frac{1}{f(x)} \times f'(x) = \frac{f'(x)}{f(x)}$

When the top is only a multiple

Often the top is the derivative times a number. Then the number just comes out in front.

For example, in $\dfrac{x}{x^2 + 1}$ :

The bottom's derivative is $2x$ .
The top is $x$ , which is $\tfrac12 \times 2x$ .

$\int \frac{x}{x^2 + 1}\,dx = \tfrac12\int \frac{2x}{x^2 + 1}\,dx = \tfrac12\ln(x^2 + 1) + C$

Only a number can be adjusted like this. If the top differs from the derivative by anything involving $x$ , the rule doesn't apply directly.

The method

Differentiate the bottom.
Compare it with the top. Is the top a number $k$ times that derivative?
If yes, write down $k$ , then the answer is $k\ln|f(x)| + C$ .
If no, you need a different method (or to split the fraction first).

Write the $k$ down before integrating. Doing it in your head is how the $\tfrac12$ goes missing.

Worked example: Example 1: the syllabus one

Find $\displaystyle\int \frac{x}{x^2 + 1}\,dx$ .

Have a go first?

Quick check

Multiple choiceQuestion 1 · Recall

Answer the 3 questions above to continue.