OCR MEI Further Pure Core 2022 June — Question 2 5 marks

Exam BoardOCR MEI
ModuleFurther Pure Core (Further Pure Core)
Year2022
SessionJune
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicIntegration using inverse trig and hyperbolic functions
TypeImproper integral to infinity with inverse trig
DifficultyChallenging +1.2 This is a Further Maths question requiring completion of the square to get (x-2)²+1, then recognizing the arctan integral form, and evaluating an improper integral to infinity. While it involves multiple steps (algebraic manipulation, inverse trig integration, and limits), each step follows standard Further Maths techniques without requiring novel insight. The improper integral aspect adds modest difficulty beyond a routine integration question.
Spec4.08c Improper integrals: infinite limits or discontinuous integrands
2 In this question you must show detailed reasoning. Find the exact value of \(\int _ { 3 } ^ { \infty } \frac { 1 } { x ^ { 2 } - 4 x + 5 } d x\)
Question 2:
AnswerMarks
2DR
 k 1  k 1
d x = d x
x 2 − 4 x + 5 ( x − 2 ) 2 + 1
3 3
= [arctan(𝑥−2)]𝑘
3
𝜋
lim[arctan(𝑘−2)] =
𝑘→∞ 2
𝜋 𝜋 𝜋
= arctan(𝑘−2)− or −
4 2 4
𝜋
⇒ integral =
AnswerMarks
4M1
A1
E1
B1
B1
AnswerMarks
[5]3.1a
1.1
2.4
1.1
AnswerMarks
2.2acompleting the square, (ignore limits)
[arctan(𝑥−2)] or [arctan𝑢] if 𝑢 = 𝑥−2 (ignore limits)
Clear limit argument
Question 2:
2 | DR
 k 1  k 1
d x = d x
x 2 − 4 x + 5 ( x − 2 ) 2 + 1
3 3
= [arctan(𝑥−2)]𝑘
3
𝜋
lim[arctan(𝑘−2)] =
𝑘→∞ 2
𝜋 𝜋 𝜋
= arctan(𝑘−2)− or −
4 2 4
𝜋
⇒ integral =
4 | M1
A1
E1
B1
B1
[5] | 3.1a
1.1
2.4
1.1
2.2a | completing the square, (ignore limits)
[arctan(𝑥−2)] or [arctan𝑢] if 𝑢 = 𝑥−2 (ignore limits)
Clear limit argument
2 In this question you must show detailed reasoning.

Find the exact value of $\int _ { 3 } ^ { \infty } \frac { 1 } { x ^ { 2 } - 4 x + 5 } d x$

\hfill \mbox{\textit{OCR MEI Further Pure Core 2022 Q2 [5]}}
This paper (15 questions)
View full paper
Q1 7 Q2 5 Q3 6 Q4 7 Q5 7 Q6 5 Q7 9 Q8 11 Q9 12 Q10 10 Q11 8 Q12 9 Q13 17 Q14 8 Q15 23