Standard +0.3 This is a straightforward substitution question with clear guidance. Students must find du = 5x^4 dx, rewrite x^9 as x^4 · x^5, change limits (u=2 to u=3), and integrate u^-3. While it requires careful algebraic manipulation across multiple steps, the method is explicitly given and follows a standard A-level integration technique with no conceptual surprises.
Question 8:
8 | Obtains 5x4
4
1 −
PI by (u−2) 5
5 | 1.1b | B1 | u = x 5 +2
du
=5x4
dx
9
x 1
∫ du
3 4
u 5x
1 3u−2
∫ du
5 2 u3
= 1 ∫ 3 u −2 −2u −3du
5 2
1 3
−u−1+u−2
5 2
11 1 1 1
= − − −
59 3 4 2
1
=
180
Substitutes for denominator and
dx operator
PI by fully correct substitution
Condone any limits or missing
integral sign or du
Condone dx in place du | 1.1a | M1
Substitutes x 5 =u−2
x=(u−2 ) 1
or 5 in at least one
place | 1.1a | M1
1 u−2
Obtains ∫ du
5 u 3
1
Condone missing or incorrect
5
or any limits
Must have du | 1.1b | A1
Integrates u−2 or u−3correctly | 1.1a | M1
1
−u−1+u−2
Obtains
5
Condone any limits | 1.1b | A1
Completes reasoned argument
by substituting correct limits
consistent with their variable to
show the given result
AG
R1 could be scored if du is
missing throughout | 2.1 | R1
Question 8 Total | 7
Q | Marking instructions | AO | Marks | Typical solution
Use the substitution $u = x^5 + 2$ to show that
$$\int_0^1 \frac{x^9}{(x^5 + 2)^3} \, dx = \frac{1}{180}$$
[7 marks]
\hfill \mbox{\textit{AQA Paper 3 2023 Q8 [7]}}