Standard +0.3 This is a straightforward integration by parts question with a standard integrand (polynomial times logarithm). The technique is routine for A-level, requiring one application of integration by parts followed by basic polynomial integration. The algebraic manipulation to express the answer in the required form is mechanical, making this slightly easier than average.
Question 4:
4 | Selects a method of integration,
which could lead to a correct
solution. Evidence of integration by
parts
OR an attempt at integration by
inspection. | AO3.1a | M1 | dv
u =ln2x; = x3
dx
du 1 x4
= ; v=
dx x 4
2
x4 x3
2
ln(2x) −∫ dx
4 1 4
1
2
x4 x4
ln(2x)−
4 16
1
24 24 1 1
= ln(4)− − ln(2)−
4 16 4 16
31 15
ln2−
4 16
31 15
so p = q = −
4 16
ALT
d 1
(x4ln2x)=4x3ln2x+x4.
dx x
2 1 x4 2
∴∫x3ln2xdx= (x4ln2x− )
4 4
1 1
24 24 1 1
= ln(4)− − ln(2)−
4 16 4 16
31 15
ln2−
4 16
31 15
p = q = −
4 16
Applies integration by parts
formula correctly
OR correctly differentiates an
expression of the form Ax4ln 2x | AO1.1b | A1
Obtains correct integral, condone
missing limits. | AO1.1b | A1
Substitutes correct limits into ‘their’
integral | AO1.1a | M1
Obtains correct p and q
FT use of incorrect integral
provided both M1 marks have
been awarded | AO1.1b | A1F
Total | 5
Marking Instructions | AO | Marks | Typical Solution
$\int_1^2 x^3 \ln(2x) dx$ can be written in the form $p\ln 2 + q$, where $p$ and $q$ are rational numbers.
Find $p$ and $q$. [5 marks]
\hfill \mbox{\textit{AQA Paper 3 Q4 [5]}}