CAIE P2 2023 March — Question 1 4 marks

Exam BoardCAIE
ModuleP2 (Pure Mathematics 2)
Year2023
SessionMarch
Marks4
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicStandard Integrals and Reverse Chain Rule
TypeUse trig identity before definite integration
DifficultyStandard +0.3 This question requires recognizing the identity tan²θ = sec²θ - 1, applying it with a chain rule adjustment for the ½x term, and evaluating definite integrals of standard forms. While it involves multiple steps and an identity that students must recall, it's a fairly standard P2 integration question with no novel problem-solving required, making it slightly easier than average.
Spec1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)

Find the exact value of \(\int_0^{\frac{\pi}{4}} 2 \tan^2(\frac{1}{2}x) \, dx\). [4]

Question 1:
AnswerMarks
1Express integrand as 2sec2 1x−2
2B1
Integrate to obtain form atan1x+bx
AnswerMarks Guidance
2M1 where ab0.
Obtain correct 4tan1x−2x
AnswerMarks
2A1
Obtain 4−πA1
4
AnswerMarks Guidance
QuestionAnswer Marks
Question 1:
1 | Express integrand as 2sec2 1x−2
2 | B1
Integrate to obtain form atan1x+bx
2 | M1 | where ab0.
Obtain correct 4tan1x−2x
2 | A1
Obtain 4−π | A1
4
Question | Answer | Marks | Guidance
Find the exact value of $\int_0^{\frac{\pi}{4}} 2 \tan^2(\frac{1}{2}x) \, dx$. [4]

\hfill \mbox{\textit{CAIE P2 2023 Q1 [4]}}