Year 12 Maths Advanced (2027) Integral calculus

The Reverse Chain Rule

20 practice questions 0 video lessons Theory + worked examples

Learn the reverse chain rule for NSW Year 12 Mathematics Advanced (NESA outcome MAV-12-05). Some integrals contain a bracketed expression together with its own derivative, and spotting that pairing lets you undo the chain rule in a single step.

You will learn to recognise integrands such as \((2x + 3)(x^2 + 3x)^4\), adjust for a missing constant, integrate composite functions using the reverse power rule and check by differentiating — a pattern-spotting skill that makes hard-looking integrals quick.

Practice 20 questions

Practice questions

Every question with a fully worked solution.

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Theory

The reverse chain rule integrates a bracket raised to a power when the inside's derivative is present. This Year 12 Mathematics Advanced topic (MAV-12-05) undoes the chain rule.

\(\displaystyle\int (ax+b)^{n}dx=\dfrac{(ax+b)^{n+1}}{a(n+1)}+C\): integrate as a power, then divide by the inner coefficient \(a\).

More generally \(\displaystyle\int f'(x)[f(x)]^{n}dx=\dfrac{[f(x)]^{n+1}}{n+1}+C\) when the derivative of the inside appears as a factor.

Integrating \((2x+1)^{n}\): power up, then divide by \(a\).

\[\int (ax+b)^{n}\,dx=\dfrac{(ax+b)^{n+1}}{a(n+1)}+C\]

\[\int f'(x)\,[f(x)]^{n}\,dx=\dfrac{[f(x)]^{n+1}}{n+1}+C\]

integral of (a x plus b) to the n is (a x plus b) to the n plus 1 over a times (n plus 1)

Method

Identify the inside and its derivative.
Raise the power by one.
Divide by the new power and by the inner coefficient.

Example 1 — Linear inside

Find \(\displaystyle\int (2x+1)^{3}\,dx\).

Solution

\(\ \)	\(=\)	\(\dfrac{(2x+1)^{4}}{4\times2}+C\)
	\(=\)	\(\dfrac{(2x+1)^{4}}{8}+C\)

Example 2 — Derivative present

Find \(\displaystyle\int 2x\,(x^{2}+1)^{4}\,dx\).

Solution

\(\ \)

\(=\)

\(\dfrac{(x^{2}+1)^{5}}{5}+C\)

Example 3 — Linear inside

Find \(\displaystyle\int (3x-2)^{5}\,dx\).

Solution

\(\ \)

\(=\)

\(\dfrac{(3x-2)^{6}}{18}+C\)

Example 4 — Apply

Find \(\displaystyle\int (5x+1)^{2}\,dx\).

Solution

\(\ \)

\(=\)

\(\dfrac{(5x+1)^{3}}{15}+C\)

Common pitfalls

Divide by the inner coefficient. \(\int(2x+1)^{3}dx=\dfrac{(2x+1)^{4}}{8}+C\).

Check the inside's derivative is present for the \(f'[f]^{n}\) form.

Differentiate to check.

Frequently asked questions

What is the reverse chain rule?

It is a way to integrate a function raised to a power by undoing the chain rule: raise the power by one and divide by the new power and the inner coefficient.

How do you integrate (2x+1) cubed?

Raise the power to 4 and divide by 4 and by the inner coefficient 2, giving (2x+1) to the fourth over 8, plus C.

When can you use the reverse chain rule directly?

When the derivative of the inside function appears as a factor, or when the inside is linear so its derivative is just a constant.

How do you check a reverse chain rule integral?

Differentiate your answer with the chain rule; it should return the original integrand.

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