AQA Further Paper 2 Specimen — Question 10 8 marks

Exam BoardAQA
ModuleFurther Paper 2 (Further Paper 2)
SessionSpecimen
Marks8
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Mark schemeDownload PDF ↗
TopicIntegration with Partial Fractions
DifficultyChallenging +1.8 This is a Further Maths question requiring partial fractions decomposition (with both quadratic and linear factors), integration of rational functions including arctangent, and proper treatment of an improper integral with limits to infinity. While the techniques are standard for Further Maths, the combination of skills, careful algebraic manipulation, and rigorous limiting process across 8 marks makes this substantially harder than average A-level questions, though not exceptionally difficult for Further Maths students who have practiced these methods.
Spec1.08h Integration by substitution4.05c Partial fractions: extended to quadratic denominators4.08c Improper integrals: infinite limits or discontinuous integrands
Evaluate the improper integral \(\int_0^{\infty} \frac{4x - 30}{(x^2 + 5)(3x + 2)} \, dx\), showing the limiting process used. Give your answer as a single term. [8 marks]
Question 10:
AnswerMarks
10Splits integrand into partial
fractions of the form
Ax B C
AnswerMarks Guidance
x2 5 3x2AO3.1a M1
 
(x2  5)(3x  2) x2 5 3x2
4x30(AxB)3x2C  x25 
Compare coefficients of x:
42A+3B
Compare coeffcients of x2:
03AC
Compare constant terms
302B+5C
302B15A
43B
302B15( )
2
B0
A2
C6
4x30 2x 6
 dx  dx
(x2 5)(3x2) x25 3x2
ln(x25)2ln(3x2)c
 4x  30
 dx
0 (x2  5)(3x  2)
a 2x 6
lim  dx
0 x25 3x2
a
a
limln(x2 5)2ln(3x2)
 
0
a
a
  x25 
limln 
(3x2)2
a 
0
  a25  5
limln ln  
a 9a212x4 4
1 5
ln   ln 
9 4
 4 
ln
 
45
Sets up an identity from which to
AnswerMarks Guidance
solve for A, B and CAO1.1a M1
Obtains correct values of
A, B and C
AnswerMarks Guidance
CAOAO1.1b A1
Integrates ‘their’ two terms
correctly
FT provided both M1 marks
AnswerMarks Guidance
awardedAO1.1b A1F
Applies the laws of logs to ‘their’
AnswerMarks Guidance
integral correctlyAO1.1a M1
Applies limits (a and 0) to ‘their’
AnswerMarks Guidance
integral correctlyAO1.1a M1
Shows the limiting process used
AnswerMarks Guidance
with clear detailed workingAO2.1 R1
Obtains correct single term solution
AnswerMarks Guidance
CAOAO1.1b A1
Total8
QMarking Instructions AO 
Question 10:
10 | Splits integrand into partial
fractions of the form
Ax B C

x2 5 3x2 | AO3.1a | M1 | 4x  30 AxB C
 
(x2  5)(3x  2) x2 5 3x2
4x30(AxB)3x2C  x25 
Compare coefficients of x:
42A+3B
Compare coeffcients of x2:
03AC
Compare constant terms
302B+5C
302B15A
43B
302B15( )
2
B0
A2
C6
4x30 2x 6
 dx  dx
(x2 5)(3x2) x25 3x2
ln(x25)2ln(3x2)c
 4x  30
 dx
0 (x2  5)(3x  2)
a 2x 6
lim  dx
0 x25 3x2
a
a
limln(x2 5)2ln(3x2)
 
0
a
a
  x25 
limln 
(3x2)2
a 
0
  a25  5
limln ln  
a 9a212x4 4
1 5
ln   ln 
9 4
 4 
ln
 
45
Sets up an identity from which to
solve for A, B and C | AO1.1a | M1
Obtains correct values of
A, B and C
CAO | AO1.1b | A1
Integrates ‘their’ two terms
correctly
FT provided both M1 marks
awarded | AO1.1b | A1F
Applies the laws of logs to ‘their’
integral correctly | AO1.1a | M1
Applies limits (a and 0) to ‘their’
integral correctly | AO1.1a | M1
Shows the limiting process used
with clear detailed working | AO2.1 | R1
Obtains correct single term solution
CAO | AO1.1b | A1
Total | 8
Q | Marking Instructions | AO | Marks | Typical Solution
Evaluate the improper integral $\int_0^{\infty} \frac{4x - 30}{(x^2 + 5)(3x + 2)} \, dx$, showing the limiting process used.

Give your answer as a single term.
[8 marks]

\hfill \mbox{\textit{AQA Further Paper 2  Q10 [8]}}
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