Definite integral — Similar modulus-and-arctan region

Question

The curves and meet at and at where solves for . Write a symmetric integral for the enclosed area and give its antiderivative form.

This practice question was inspired by Question 13(c) in the 2021 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.By evenness, area .
State the final answer clearly in the form the question requested.Primitive , so area .

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

Verification: Same symmetry-plus-arctan structure as the exam.

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