Reverse chain rule
Objective. Recognise derivative-over-function and composite patterns.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
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Learn: the key idea
The reverse chain rule integrates expressions where the integrand is a function's derivative multiplied by (or divided into) a composite of that function, undoing the chain rule. If the integrand has the form f′(x)/f(x), the integral is ln|f(x)| + c; if it is f′(x)[f(x)]ⁿ, the integral is [f(x)]^(n+1)/(n+1) + c. For example, ∫ 2x cos(x²) dx = sin(x²) + c, since 2x is the derivative of x². Always check that the derivative of the inner function is present as a factor before applying this shortcut.
Worked example
Use the reverse chain rule to find ∫ 6x²(x³ + 1)⁴ dx.
- 1Notice 6x² is proportional to the derivative of x³ + 1, which is 3x²
- 2Write the integral as 2∫ 3x²(x³+1)⁴ dx
- 3Integrate using [f(x)]⁵/5: 2 × (x³+1)⁵/5
- 4Simplify to (2/5)(x³+1)⁵ + c
Answer: (2/5)(x³ + 1)⁵ + c
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Practice
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Start practisingCommon mistake. A common error is applying the reverse chain rule when the outer derivative factor is missing or has the wrong constant, such as integrating ∫x cos(x²)dx as sin(x²) without inserting the compensating ½. Always check the coefficient matches the true derivative exactly.
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