ACARA ACMMM117, Linearity of anti-differentiation

Year 11-12 · Mathematical Methods

Recognise and use the linearity of anti-differentiation to integrate sums of terms and constant multiples of functions.

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Teaching guide

What students learn

Recognise and use the linearity of anti-differentiation to integrate sums of terms and constant multiples of functions.

How to teach it

Anchor it as differentiation run backwards: since d/dx(x^(n+1)) = (n+1)x^n, reversing gives integral of x^n dx = x^(n+1)/(n+1). Raise the power by one, divide by the new power, then always add + C. Check each answer by differentiating it back to the original.

Worked example

Find the indefinite integral (antiderivative) of f(x) = -8x^4 with respect to x. Include the constant of integration + C.

Answer: -(8/5)x^5 + C

Why: Power rule (integral of a x^n dx = a/(n+1) x^(n+1)), term by term: -8x^4 -> -8/5 x^5 = -(8/5)x^5. So F(x) = -(8/5)x^5 + C.

Generated and checked in code, so it is correct.

Watch out for

Forgetting + C, dividing by the old power instead of the new (n+1), and trying to apply the rule to 1/x (n = -1), which the power rule excludes.

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More Year 11-12 Mathematical Methods codes: ACMMM114 · ACMMM116 · ACMMM121 · ACMMM122 · ACMMM125 · ACMMM131 · ACMMM132

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Australian Curriculum content descriptions (c) ACARA, used under CC BY 4.0. Descriptions are paraphrased. See australiancurriculum.edu.au.