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WASpecialist MathematicsSyllabus dot point

How does reversing the chain rule let us integrate functions built by composition?

Integrate by substitution, changing the variable and the limits to evaluate definite and indefinite integrals

WACE Specialist Unit 4 integration by substitution: choosing u, replacing dx with du over the derivative, changing limits for definite integrals, and the reverse chain rule, with a worked example.

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  1. What this dot point is asking
  2. The method
  3. Choosing the substitution
  4. Definite integrals: change the limits

What this dot point is asking

SCSA wants confident use of substitution for both indefinite and definite integrals, including recognising when the integrand contains a function and (a multiple of) its derivative.

The method

Choose u=g(x)u = g(x) so that the integrand simplifies. Differentiate to get du=g′(x) dxdu = g'(x)\,dx, and replace every xx-expression, including dxdx, by its uu-equivalent. The integral must end up entirely in uu with no stray xx remaining. Integrate, then for an indefinite integral substitute xx back.

Choosing the substitution

Look for an inner function whose derivative is also present (up to a constant factor). Common choices: the expression under a root, the exponent of ee, the argument of a trig function, or the denominator. If the derivative is present only up to a constant, adjust by that constant.

Definite integrals: change the limits

This is cleaner and avoids errors from reverting. If you do keep xx-limits, you must back-substitute before applying them.