Definite integral — A cosine-square integral

Question

Evaluate .

This practice question was inspired by Question 11(g) in the 2025 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: find a suitable antiderivative, substitute the upper and lower bounds, and subtract in that order..
Simplify the previous line carefully, keeping exact values where possible.An antiderivative is .
Finish the calculation, then check that the result meets the question’s conditions.Both sine boundary terms vanish; the result is .
State the final answer clearly in the form the question requested.Final answer:

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

Verification: The integrand completes half a period and has average value .

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