Mathematics · May / June 2021 · Paper 41 · Question 4
YMA01
As printed in the paper
The question as text
4. Use algebraic integration and the substitution u = x to find the exact value of 4 10 ∫ d x 5 x + 2 x x 1 Write your answer in the form 4ln a , where a and b are integers to be found. b (Solutions relying entirely on calculator technology are not acceptable.) (8)
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Show the answer and mark scheme
4
∫
10
d
x
5
x
+
2
x x
1
d
x
u
=
x
⇒
x
=
u
2
⇒
=
2
u
d
u
∫
∫
10
10
d
x
=
2 d
u u
5
x
+
2
x x
2
3
5
u
+
2
u
∫
20
∫
4
8
d
d
=
u
=
−
u
2
u
5
+
2
u
5
2
u
+
u
=
4ln
u
−
4ln(5
+
2 )
u
Limits [
]
2
=
4ln
u
−
4ln(5
+
2 )
u
=
4ln 2
−
4ln9
+
4ln 7
=
...
1
14
=
4ln
9
B1: For d
x
d
u
1
1
−
u =
2
u
or
=
x
2
or equivalent
d
d
2
x
M1: Attempts to write all terms in the integral in terms of u (inc the dx)
Condone slips for this mark BUT the dx CANNOT just be replaced by du.
Look for either
x
→
u
2
or
x x
→
u
3
with either d
x
→
f ( )d
u
u
or d
x
→
k
×
d ,
u k
≠
1
∫
10
A1: Correct integrand in terms of just u which may be unsimplified. E.g
2 d
2
3
5
u
+
2
u
dM1: Attempts to use PF and writes their integrand in terms of its component fractions.
∫
P
∫
p
q
Look for an integral of the form
2 d
u
→
+
d
u
u
Q
+
Ru
Qu
+
Ru
∫
4
8
A1: Correct PF
(
d
)
−
u
u
5
+
2
u
ddM1: For ...ln
u
±
...ln(5
+
2 )
u
but follow through on their PF's which must be of a similar form
M1: Uses the limits 1 and 2 within their attempted integral. The integration may be incorrect.
Alternatively substitutes u
=
x
and uses the limits 1 and 4 within their attempted integral
14
A1:
4ln
9
d
y
d
y
−
2
x
−
2
x
2
y
e
2 e
y
3
=
−
−
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