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11.4PureStretch

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.

26 min Integration
Edexcel A level Maths: 11.4 Reverse Chain Rule (Integration)Curated by Kepler, not produced by us. Watch it before 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
1

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.

∫f′(x)f(x) dx=ln⁡∣f(x)∣+c\int \frac{f'(x)}{f(x)}\,dx = \ln|f(x)| + c
Log pattern.
∫f′(x) [f(x)]n dx=[f(x)]n+1n+1+c\int f'(x)\,[f(x)]^n\,dx = \frac{[f(x)]^{n+1}}{n+1} + c
Power pattern.
2

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 ∫2xx2+1 dx\int \dfrac{2x}{x^2 + 1}\,dx.
1Numerator 2x2x is the derivative of x2+1x^2 + 1.
2=ln⁡∣x2+1∣+c= \ln|x^2 + 1| + c.
Answer
ln⁡(x2+1)+c\ln(x^2 + 1) + c

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

Formula recap

∫f′(x)f(x) dx=ln⁡∣f(x)∣+c\int \frac{f'(x)}{f(x)}\,dx = \ln|f(x)| + c
Log pattern.
∫f′(x)[f(x)]n dx=[f(x)]n+1n+1+c\int f'(x)[f(x)]^n\,dx = \frac{[f(x)]^{n+1}}{n+1} + c
Power pattern.

Common mistakes to avoid

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 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.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. ∫f′(x)f(x) dx\int \frac{f'(x)}{f(x)}\,dx equals:

    • ln⁡∣f(x)∣+c\ln|f(x)| + c
    • f(x)2+cf(x)^2 + c
    • 1f(x)+c\frac{1}{f(x)} + c
    • f′(x)+cf'(x) + c
    Show answer and explanation

    Answer: ln⁡∣f(x)∣+c\ln|f(x)| + c

    Log pattern.

  2. ∫2xx2+1 dx\int \frac{2x}{x^2+1}\,dx equals:

    • ln⁡(x2+1)+c\ln(x^2+1) + c
    • 1x2+1+c\frac{1}{x^2+1}+c
    • 2ln⁡(x2+1)+c2\ln(x^2+1)+c
    • x2+1+cx^2+1+c
    Show answer and explanation

    Answer: ln⁡(x2+1)+c\ln(x^2+1) + c

    Top = derivative of bottom.

  3. ∫f′(x)[f(x)]n dx\int f'(x)[f(x)]^n\,dx equals:

    • [f(x)]n+1n+1+c\frac{[f(x)]^{n+1}}{n+1}+c
    • [f(x)]n+c[f(x)]^n + c
    • ln⁡∣f(x)∣+c\ln|f(x)|+c
    • n[f(x)]n−1+cn[f(x)]^{n-1}+c
    Show answer and explanation

    Answer: [f(x)]n+1n+1+c\frac{[f(x)]^{n+1}}{n+1}+c

    Power pattern.

  4. If the derivative factor is out by a constant k, you:

    • divide by k
    • multiply by k
    • ignore it
    • add k
    Show answer and explanation

    Answer: divide by k

    Adjust the constant.

  5. The reverse chain rule reverses which rule?

    • the chain rule
    • the product rule
    • the quotient rule
    • integration by parts
    Show answer and explanation

    Answer: the chain rule

    Hence the name.

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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