The reverse chain rule integrates expressions where a function appears together with its derivative. The reliable method is to consider a likely answer, differentiate it, and adjust by a constant so it matches the integrand. Key patterns are ∫ g′(x)e^(g(x)) dx = e^(g(x)) + c and ∫ f′(x)/f(x) dx = ln|f(x)| + c.
Reverse chain rule patterns
- ∫ g′(x)[g(x)]ⁿ dx = [g(x)]ⁿ⁺¹/(n + 1) + c
- ∫ g′(x)e^(g(x)) dx = e^(g(x)) + c
- ∫ f′(x)/f(x) dx = ln|f(x)| + c
- ∫ tan x dx = −ln|cos x| + c = ln|sec x| + c
Method: consider a likely answer, differentiate it, then adjust by a constant.
When does the reverse chain rule apply?
Look for a function and its derivative appearing together, possibly up to a constant multiple. In 3x²(x³ + 2)⁴, the inner function is x³ + 2 and its derivative, 3x², is sitting outside. That is the signal. If the derivative of the inside is not present, even approximately, this method will not work.
Consider, differentiate, adjust
Guess the form of the answer from the outside function: raise the power by one, or keep the exponential, or take a logarithm. Differentiate your guess using the chain rule, compare the result with the integrand, and multiply by whatever constant makes them match.
Always differentiate your guess to find the constant. Forgetting the adjustment factor is the most common error, and the check takes seconds.
Exponentials with a function in the power
For g′(x)e^(g(x)), consider e^(g(x)). Differentiating brings the inner derivative down, so it matches when g′(x) is present. If only a constant multiple of g′(x) is present, adjust by that constant.
The f′(x)/f(x) pattern
When the numerator is the derivative of the denominator, or a constant multiple of it, the integral is a natural logarithm: ∫ f′(x)/f(x) dx = ln|f(x)| + c. Check the pattern first: is the top the derivative of the bottom? If it is twice the derivative, the answer is 2 ln|f(x)|; if it is half, ½ ln|f(x)|.
Trig and the reverse chain rule
The same ideas work for trig functions. A power of sin x times cos x, or a power of cos x times sin x, can be integrated by considering the next power up. Writing tan x = sin x / cos x shows that the top is minus the derivative of the bottom, so ∫ tan x dx = −ln|cos x| + c, which is the same as ln|sec x| + c.
Worked examples
Example 1. Find ∫ 3x²(x³ + 2)⁴ dx.
- Consider (x³ + 2)⁵.
- Differentiate: 5(x³ + 2)⁴ × 3x² = 15x²(x³ + 2)⁴.
- The integrand is ⅕ of this, so multiply by ⅕.
Answer: ⅕(x³ + 2)⁵ + c.
Marks: M1 for considering and differentiating, M1 for adjusting by ⅕, A1 for the answer.
Example 2. Find ∫ 3x²e^(x³) dx.
- Consider e^(x³).
- Differentiate: 3x²e^(x³), which matches exactly.
Answer: e^(x³) + c.
Marks: M1 for considering e^(x³), A1 for the answer.
Example 3. Find ∫ 4x³/(x⁴ + 1) dx.
- The derivative of x⁴ + 1 is 4x³, exactly the numerator.
- Use ∫ f′(x)/f(x) dx = ln|f(x)| + c.
Answer: ln(x⁴ + 1) + c. The modulus can be dropped because x⁴ + 1 is always positive.
Marks: M1 for recognising f′(x)/f(x), A1 for the answer.
Example 4. Find ∫ cos³x sin x dx.
- Consider cos⁴x.
- Differentiate: 4cos³x × (−sin x) = −4cos³x sin x.
- The integrand is −¼ of this.
Answer: −¼cos⁴x + c.
Marks: M1 for differentiating cos⁴x, A1 for the answer.
Example 5. Find the exact value of ∫ 2xe^(x²) dx from 0 to 1.
- Consider e^(x²): its derivative is 2xe^(x²), which matches.
- Evaluate [e^(x²)] from 0 to 1: e¹ − e⁰.
Answer: e − 1.
Marks: M1 for [e^(x²)], A1 for e − 1.
Common mistakes
- Forgetting to adjust the constant. Differentiate your guess and compare; the adjustment factor is often a fraction.
- Using the method without the inner derivative. ∫ e^(x²) dx cannot be done this way, because there is no factor of x.
- Missing the f′(x)/f(x) pattern. If the top is a multiple of the derivative of the bottom, the answer is a logarithm.
- Sign errors with cos. The derivative of cos x is −sin x, which introduces a minus sign in the adjustment.
- Dividing by a non-constant. You may only adjust by a constant, never by a function of x.
How the reverse chain rule is examined
The reverse chain rule belongs to the Pure 3 paper (WMA13) in Edexcel International A-Level, and to second-year Pure in Edexcel A-Level (9MA0). Questions commonly:
- ask you to show that an integral equals a given result, often a logarithm
- integrate a power or exponential of a function multiplied by its derivative
- use the result to find an exact area
- use the f′(x)/f(x) pattern after spotting a hidden derivative
Show-that and area questions with full mark schemes are in the Pure 3 integration exam questions. The integration worksheet and the A-Level integration revision engine give further practice.
This reverses the chain rule. For harder cases where the derivative is not simply a constant multiple, use integration by substitution.
Frequently asked questions
What is the reverse chain rule?
A method for integrating a function multiplied by the derivative of its inside: guess the answer, differentiate it, and adjust by a constant.
What is the integral of f′(x)/f(x)?
ln|f(x)| + c.
What is the integral of tan x?
−ln|cos x| + c, which can also be written ln|sec x| + c.