OCR MEI Further Pure Core 2022 June — Question 7 9 marks

Exam BoardOCR MEI
ModuleFurther Pure Core (Further Pure Core)
Year2022
SessionJune
Marks9
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicIntegration with Partial Fractions
TypePartial fractions with irreducible quadratic
DifficultyChallenging +1.2 This is a Further Maths partial fractions question requiring decomposition with an irreducible quadratic, followed by integration using both logarithmic and arctangent forms. While it involves multiple techniques (partial fractions setup, solving for constants, integrating ln and arctan terms, evaluating definite integral), the structure is standard and the arithmetic works out cleanly to the given answer. It's harder than typical A-level questions due to the irreducible quadratic, but follows a well-practiced procedure for Further Maths students.
Spec4.08f Integrate using partial fractions
7 In this question you must show detailed reasoning.
Show that \(\int _ { 2 } ^ { 3 } \frac { x + 1 } { ( x - 1 ) \left( x ^ { 2 } + 1 \right) } d x = \frac { 1 } { 2 } \ln 2\).
Question 7:
AnswerMarks
7𝑥+1 𝐴 𝐵𝑥+𝐶
= +
(𝑥−1)(𝑥2+1) 𝑥−1 𝑥2+1
⇒ 𝑥+1 = 𝐴(𝑥2+1)+(𝐵𝑥+𝐶)(𝑥−1)
𝑥 = 1 ⇒ 2 = 2𝐴 ⇒ 𝐴 = 1
coefficient of 𝑥2: 0 = 𝐴+𝐵 ⇒ 𝐵 = −1
constants: 1 = 𝐴−𝐶 ⇒ 𝐶 = 0
 3 x + 1  3 ( 1 x )
d x = − d x
( x − 1 ) ( x 2 + 1 ) x − 1 x 2 + 1
2 2
1 3
= [ln(𝑥−1)− 𝑙𝑛(𝑥2+1)]
2
2
1 1
= ln2− ln10+ ln5 [−ln1]
2 2
1 10
= ln2− ln
2 5
1 1
= ln2− ln2 = ln2
AnswerMarks
2 2M1
A1
A1
A1
B1ft
M1
A1ft
M1
A1
AnswerMarks
[9]2.1
1.1
2.1
2.1
2.1
2.1
1.1
1.1
AnswerMarks
2.2acorrect partial fractions
ln(𝑥−1)
𝑘ln(𝑥2+1) or u = x 2 + 1   1 d u
2 u
1 1
𝑘 = − or − ln𝑢
2 2
Combining two of their logarithm terms correctly
AG
Question 7:
7 | 𝑥+1 𝐴 𝐵𝑥+𝐶
= +
(𝑥−1)(𝑥2+1) 𝑥−1 𝑥2+1
⇒ 𝑥+1 = 𝐴(𝑥2+1)+(𝐵𝑥+𝐶)(𝑥−1)
𝑥 = 1 ⇒ 2 = 2𝐴 ⇒ 𝐴 = 1
coefficient of 𝑥2: 0 = 𝐴+𝐵 ⇒ 𝐵 = −1
constants: 1 = 𝐴−𝐶 ⇒ 𝐶 = 0
 3 x + 1  3 ( 1 x )
d x = − d x
( x − 1 ) ( x 2 + 1 ) x − 1 x 2 + 1
2 2
1 3
= [ln(𝑥−1)− 𝑙𝑛(𝑥2+1)]
2
2
1 1
= ln2− ln10+ ln5 [−ln1]
2 2
1 10
= ln2− ln
2 5
1 1
= ln2− ln2 = ln2
2 2 | M1
A1
A1
A1
B1ft
M1
A1ft
M1
A1
[9] | 2.1
1.1
2.1
2.1
2.1
2.1
1.1
1.1
2.2a | correct partial fractions
ln(𝑥−1)
𝑘ln(𝑥2+1) or u = x 2 + 1   1 d u
2 u
1 1
𝑘 = − or − ln𝑢
2 2
Combining two of their logarithm terms correctly
AG
7 In this question you must show detailed reasoning.\\
Show that $\int _ { 2 } ^ { 3 } \frac { x + 1 } { ( x - 1 ) \left( x ^ { 2 } + 1 \right) } d x = \frac { 1 } { 2 } \ln 2$.

\hfill \mbox{\textit{OCR MEI Further Pure Core 2022 Q7 [9]}}
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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