Antidifferentiation — Same structure different shift

Question

Using , find .

This practice question was inspired by Question 11(c) in the 2024 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., .
Simplify the previous line carefully, keeping exact values where possible.Integral .
Finish the calculation, then check that the result meets the question’s conditions.Antiderivative .
State the final answer clearly in the form the question requested.Answer: .

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

Verification: Differentiate to recover .

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