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.
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.
Log pattern.
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.
1Numerator is the derivative of .
2.
Answer
Tip — Check whether the “top” is the derivative of the “bottom” (up to a constant).
Formula recap
Log pattern.
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.