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 Video by Zeeshan Zamurred Integration

Edexcel A level Maths: 11.4 Reverse Chain Rule (Integration)Watch the full walkthrough before the notes below.

Open on YouTube

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 )} d x = ln ∣ f (x) ∣ + c

Log pattern.

\int f^{'} (x) [f (x)]^{n} d x = \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.

1Numerator

2 x

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).

Formula recap

\int \frac{f ^{'} ( x )}{f ( x )} d x = ln ∣ f (x) ∣ + c

Log pattern.

\int f^{'} (x) [f (x)]^{n} d x = \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

Ready to lock in The Reverse Chain Rule? Pick a mode and earn XP & Dobloons.