2022 HSC Mathematics Extension 1
Question 11(b):
Antidifferentiation
Find the exact value of using the substitution .
How to recognise this question
Question type: antidifferentiation
- The command and notation point to antidifferentiation.
- The requested response is a worked solution.
How to handle it: choose a substitution or identity that simplifies the integrand, then reverse-differentiate to check.
Watch out: Do not select a formula until its domain, interval, sign and units match the question.
Step-by-step answer
- Let , so and .
- Limits: ; .
- Integral becomes .
- NESA: 3 marks correct; 2 for correct integrand after substitution (or equivalent); 1 for attempting the substitution.
Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.
The paper records 3 total marks for this group. Where the official marking guide provides useful partial-credit criteria, those criteria are stated in the steps above.
Try your hand at it again by answering these questions
Question 1
Evaluate using .
Question 2
Evaluate via .
Question 3
With , show that , then apply limits to .