The Reverse Chain Rule is an integration technique used to find the antiderivative of a composite function (i.e., a function inside another function) when it is multiplied by the derivative of the inner function. It is essentially the Chain Rule for differentiation applied backwards.
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Reverse Chain Rule
Integration by substitution is a broader technique. For simpler integrals, you might use the Reverse Chain Rule “by inspection” by mentally keeping track of the inner function’s derivative. For more complex integrals, u-substitution provides a rigorous way to swap variables and balance the equation so you don’t make mistakes.
The formula for the reverse chain rule is as follows:
![\int f'(x)[f(x)]^n \,dx = \frac{[f(x)]^{n+1}}{n+1} + C](https://s0.wp.com/latex.php?latex=%5Cint+f%27%28x%29%5Bf%28x%29%5D%5En+%5C%2Cdx+%3D+%5Cfrac%7B%5Bf%28x%29%5D%5E%7Bn%2B1%7D%7D%7Bn%2B1%7D+%2B+C&bg=ffffff&fg=666666&s=3&c=20201002)
How to Apply
- Identify the inner function: Find the piece inside another function or raised to a power.
- Check for the derivative: Verify that the derivative of this inner function is multiplied on the outside.
- Integrate: Raise the power of the composite function by 1, divide by the new power, and add the constant of integration C. (Adjust for any missing constants by multiplying/dividing by that number.)
For example, to integrate ∫ 2x cos(x2) dx, you can:
- Inner Function: f(x) = x2
- Derivative: The derivative of x2 is 2x. Because 2x is multiplied right next to cos, we can apply the rule.
- Integrate: The antiderivative of cos is sin, leaving the inner function untouched. Result: sin(x2) + C.
Is the Reverse Chain Rule the Same as u-substitution?
They are essentially the same concept, but with a difference in formality. The Reverse Chain Rule is the conceptual principle of integrating composite functions, while u-substitution is the formal, step-by-step mathematical technique used to execute it.
For simpler integrals, you might use the Reverse Chain Rule “by inspection” by mentally keeping track of the inner function’s derivative. For more complex integrals, u-substitution provides a rigorous way to swap variables and balance the equation so you don’t make mistakes.
For example, to integrate ∫ 2x cos(x2) dx, you can:
- Reverse Chain Rule: Recognize that 2x is the derivative of the “inside” function x2. Since the derivative of sin(u) is cos(u), you know the antiderivative is sin(x2) + C.
- U-substitution: Let u = x2, which makes the derivative du = 2x dx. Substituting this yields ∫ cos(u) du, which resolves to sin(u) + C before you substitute x2 back in.
Case Where U-Substitution Is Mandatory Because the Reverse Chain Rule Is Too Hard to Do Mentally
Take the integral: ∫ x2 cos(x3) dx
While mental math works for linear inner functions, this cubic inner function requires formal u-substitution. Because the derivative of x3 is a multiple of x2, this integral can be evaluated by letting u equal the inner function.
- Assign u and find du
- Let u = x3
- Differentiate u with respect to x: du = 3x2 dx
- Rearrange to match the leftover term in the integral: ⅓ du = x2 dx
- Rewrite the integral: Substitute u and du into the equation, which greatly simplifies it: ∫ cos(u) (⅓ du)
- Integrate: Pull out the constant ⅓ and take the antiderivative of cos(u), which is sin(u): ⅓ sin(u) + C
- Substitute back: Replace u with x3 to get the final answer: ⅓ sin(x3) + C
The mental “Reverse Chain Rule” fails here because you would need to look at x2 cos(x3) and instantly guess the exact function whose derivative equals this integrand. Because of the non-linear “inner function” x3, its derivative 3x2 leaves behind a variable component. Balancing this “leftover” variable component and the chained function is exactly what the u-substitution algorithm forces you to align.
Is Integration by Substitution Always the Reverse Chain Rule?
The Reverse Chain Rule strictly applies to integrals that are in the form ∫ f(g(x))g'(x) dx, where the derivative of the “inside” function is already present. Integration by substitution is a broader technique. While it is fundamentally justified by the chain rule, you can also use it to algebraically manipulate an integrand into an entirely different form. Advanced techniques like trigonometric substitution use this principle to leverage algebraic identities rather than simply reverse a derivative.
Comparison Summary
| Feature | Reverse Chain Rule | U-Substitution |
|---|---|---|
| Execution | Done mentally by inspection | Done on paper by changing variables |
| Speed | Very fast for simple problems | Slower, but highly structured |
| Best For | Linear insides or obvious derivatives | Complex fractional or radical expressions |
| Error Risk | High (easy to drop a constant factor) | Low (the algebra forces constants to balance) |
References
Paul. “Reverse Chain Rule.” Save My Exams. Accessed July 20, 2026. https://www.savemyexams.com/igcse/further-maths/cie/additional-maths/25/revision-notes/calculus/integration/reverse-chain-rule/.
Paul. “Reverse Chain Rule – A Level Maths Revision Notes.” Save My Exams. Accessed July 20, 2026. https://www.savemyexams.com/a-level/maths/edexcel/18/pure/revision-notes/integration/further-integration/reverse-chain-rule/.
“The Reverse Chain Rule.” HSCprep, May 11, 2024. https://hscprep.com.au/hsc-maths-advanced/the-reverse-chain-rule/.
Additional Reading
“7.02 The Reverse Chain Rule.” mathspace, Accessed July 20, 2026. https://mathspace.co/textbooks/syllabuses/Syllabus-826/topics/Topic-18357/subtopics/Subtopic-249868/?activeTab=theory.
“Chain Rule.” Mathematical Mysteries, June 11, 2026. https://mathematicalmysteries.org/chain-rule/.
“The Chain Rule.” web.ma.utexas.edu, Accessed July 20, 2026. https://web.ma.utexas.edu/users/m408n/AS/LM3-4-2.html.
Videos
Khan Academy demonstrates how to identify patterns in integrals that match the derivative of a composite function. By reversing the chain rule, you can solve these problems mentally, providing a faster alternative to traditional substitution methods.
Khan Academy demonstrates how to evaluate an indefinite integral involving a sine function by applying the reverse chain rule. The tutorial explains how to identify a function and its derivative within the integrand to simplify the integration process without explicitly using u-substitution.
First we show how to use the reverse chain rule. We recognize that we have a function composition right next to the derivative of the interior function in our integrand. At least it’s the variable parts of the derivative of the interior function, and it’s not a big deal to take care of the constants by multiplying the integrand by a constant and compensating out in front.
Now that we have a function composition right next to the derivative of the interior function, we apply the chain rule backwards and quickly find the antiderivative of x^2(2-x^3)^100, and we’re done!
Now we integrate x^2*(2-x^3)^100 using a formal u-substitution. We still need essentially the derivative of the interior function in the integrand, otherwise the u-sub won’t be fruitful, and those variable pieces (x^2) are just right to miss being the derivative of the interior function by only a constant multiple.
So we let u=2-x^3, du=-3x^2*dx, solve for dx and sub everything into the integral to transform the integral in terms of u. After finding the antiderivative in terms of u, we have to substitute for u in terms of x to arrive at the same final answer, and we’re done.
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