Exam Board AQA Module Paper 2 (Paper 2) Year 2020 Session June Marks 6 Paper Download PDF ↗ Mark scheme Download PDF ↗ Topic Integration by Substitution Difficulty Standard +0.3 This is a straightforward integration by substitution question with clear structure: the integrand suggests u = 4x + 1, requiring standard algebraic manipulation to express x in terms of u, adjusting limits, and integrating polynomial terms. While it requires careful bookkeeping across multiple steps for 6 marks, it follows a completely standard template with no conceptual surprises, making it slightly easier than average. Spec 1.08h Integration by substitution
Use integration by substitution to show that
$$\int_{-\frac{3}{4}}^6 x\sqrt{4x + 1} \, dx = \frac{875}{12}$$
Fully justify your answer.
[6 marks]
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Question 5:
Answer Marks
Guidance
5 Uses a suitable substitution
u =4x+1 oru = 4x+1OE 3.1a
M1
u =4x+1⇒ =4
dx
1
x=− ⇒u =0
4
x=6⇒u =25
u−1
x=
4
6 25u−1 1
∫ ∫
x 4x+1 dx= u du
1 4 4
− 0
4
1 25 3 1
∫
= u2 −u2du
16
1
25
5 3
1 2u2 2u2
= −
16 5 3
0
5 3
1252 252
= −
8 5 3
875
=
12
Differentiates their substitution
Answer Marks
Guidance
correctly 1.1b
A1
Completes substitution to obtain
correct integrand for their suitable
Answer Marks
Guidance
substitution. Can be unsimplified. 1.1a
M1
Correctly integrates their simplified
integrand provided it is of the form
3 1
Au2 −u2 or B ( u4 −u2)
Answer Marks
Guidance
1.1a
A1
Substitutes correct limits for their
Answer Marks
Guidance
substitution or 6 and -1/4 for x 1.1a
M1
Completes rigorous argument to
show the required result.
Answer Marks
Guidance
AG 2.1
A1
Total 6
Q Marking instructions
AO
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Question 5:
5 | Uses a suitable substitution
u =4x+1 oru = 4x+1OE | 3.1a | M1 | du
u =4x+1⇒ =4
dx
1
x=− ⇒u =0
4
x=6⇒u =25
u−1
x=
4
6 25u−1 1
∫ ∫
x 4x+1 dx= u du
1 4 4
− 0
4
1 25 3 1
∫
= u2 −u2du
16
1
25
5 3
1 2u2 2u2
= −
16 5 3
0
5 3
1252 252
= −
8 5 3
875
=
12
Differentiates their substitution
correctly | 1.1b | A1
Completes substitution to obtain
correct integrand for their suitable
substitution. Can be unsimplified. | 1.1a | M1
Correctly integrates their simplified
integrand provided it is of the form
3 1
Au2 −u2 or B ( u4 −u2)
| 1.1a | A1
Substitutes correct limits for their
substitution or 6 and -1/4 for x | 1.1a | M1
Completes rigorous argument to
show the required result.
AG | 2.1 | A1
Total | 6
Q | Marking instructions | AO | Marks | Typical solution
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Use integration by substitution to show that
$$\int_{-\frac{3}{4}}^6 x\sqrt{4x + 1} \, dx = \frac{875}{12}$$
Fully justify your answer.
[6 marks]
\hfill \mbox{\textit{AQA Paper 2 2020 Q5 [6]}}