1.08hPureStretch

The Reverse Chain Rule

The reverse chain rule integrates expressions where one factor is (almost) the derivative of another. Two key patterns let you write the answer down quickly — and adjust by a constant.

30 min Integration

The big picture

The derivative of $\ln|f(x)|$ is $f'(x)/f(x)$ wherever f is nonzero. Compare the numerator with the denominator's derivative before choosing a more complicated method. For $f'(x)[f(x)]^n$ , the antiderivative is $[f(x)]^{n+1}/(n+1)$ when $n\neq-1$ . If the numerator is only a constant multiple of $f'(x)$ , adjust that constant outside the integral. A near match is not enough: the numerator must be the needed derivative times a constant. If an extra variable factor remains, a different substitution or another integration method may be needed. Differentiate the proposed antiderivative to verify the exact integrand.

Edexcel A level Maths: 11.4 Reverse Chain Rule (Integration)Curated by Kepler, not produced by us. Watch it, then work through the notes below.

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What you'll be able to do

Recognise the reverse chain rule patterns
Integrate f′(x)/f(x) to ln|f(x)|
Integrate f′(x)[f(x)]ⁿ
Adjust by a constant factor

The two patterns

When the numerator is the derivative of the denominator, the integral is a logarithm. When a power of a function is multiplied by its derivative, raise the power.

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

Log pattern.

\int f'(x)\,[f(x)]^n\,dx = \frac{[f(x)]^{n+1}}{n+1} + c

Power pattern.

Adjusting the constant

If the derivative factor is out by a constant multiple, integrate as if it matched and divide by that constant.

Example 1Find

\int \dfrac{2x}{x^2 + 1}\,dx

1Numerator

2x

is the derivative of

x^2 + 1

= \ln|x^2 + 1| + c

Answer

\ln(x^2 + 1) + c

Tip — Check whether the “top” is the derivative of the “bottom” (up to a constant).

Recognise a derivative divided by its function

The derivative of $\ln|f(x)|$ is $f'(x)/f(x)$ wherever f is nonzero. Compare the numerator with the denominator's derivative before choosing a more complicated method.

For $f'(x)[f(x)]^n$ , the antiderivative is $[f(x)]^{n+1}/(n+1)$ when $n\neq-1$ . If the numerator is only a constant multiple of $f'(x)$ , adjust that constant outside the integral.

A near match is not enough: the numerator must be the needed derivative times a constant. If an extra variable factor remains, a different substitution or another integration method may be needed. Differentiate the proposed antiderivative to verify the exact integrand.

Example 1Find

\int\dfrac{3x}{x^2+4}\,dx

1The derivative of

x^2+4

2x

2Rewrite

3x=(3/2)(2x)

3The integral is

(3/2)\ln(x^2+4)+C

4Here

x^2+4>0

, so absolute-value notation is optional.

Answer

\tfrac32\ln(x^2+4)+C

Think like an examiner

Write the defining equations and show the substitution before evaluating.
Check the domain, units and interpretation of the final result.

Common misconceptions

Using the log pattern when the top is not the derivative of the bottom.

Verify (up to a constant) that f′ matches; else use substitution.

Forgetting to divide by the adjusting constant.

Scale by 1/k if the derivative factor is out by k.

Key results to remember

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

Log pattern.

\int f'(x)[f(x)]^n\,dx = \frac{[f(x)]^{n+1}}{n+1} + c

Power pattern.

Key takeaways

∫f′/f = ln|f| + c.
∫f′·fⁿ = fⁿ⁺¹/(n+1) + c.
Adjust by a constant if the derivative factor is scaled.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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