Variation 2 · answer guide

Trigonometric substitution — A tangent substitution

Question

Using , evaluate .

This practice question was inspired by Question 3 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: write the substitution, its derivative and the new limits before simplifying.
Simplify the previous line carefully, keeping exact values where possible. and the new upper bound is .
Finish the calculation, then check that the result meets the question’s conditions.
Match the result to the choices and state the correct option.Therefore option B is correct:

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

Verification: Differentiate .

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