Antidifferentiation — Half-factor check

Question

With , show that , then apply limits to .

This practice question was inspired by Question 11(b) in the 2022 NSW HSC Mathematics Extension 1 examination, © NESA. It is an original TestMum variation that practises the same mathematical idea, not an official NESA question.

Set up the solution: choose a substitution or identity that simplifies the integrand, then reverse-differentiate to check.Chain rule: derivative of is .
State the final answer clearly in the form the question requested.Definite integral .

Common mistake: changing so many features that the new example no longer tests the same mathematical idea.

Verification: Recognising the chain-rule reverse is equivalent to the -sub.

Any mark labels are a TestMum study aid, not an official NESA marking allocation.