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Past exam of the mathematics course of the University of Cambridge
/
2021
/
ib
/
Paper 1
/
3B
/
Solution
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Mathematics course of the University of Cambridge
Past exam of the mathematics course of the University of Cambridge
2021
ib
Paper 1
3B
Created
2026-09-24
Updated
2026-09-25
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The integrand has an order-two
pole
at
0
and
a
simple
pole
at
2
. Write it near zero
as
h
(
z
)
/
z
2
, where
h
(
z
)
=
z
−
2
z
2
+
e
z
.
(1)
Its
residue
at zero is
h
′
(
0
)
=
(
z
−
2
)
2
(
2
z
+
e
z
)
(
z
−
2
)
−
(
z
2
+
e
z
)
z
=
0
=
−
4
3
.
(2)
At
z
=
2
the
residue
is
2
2
2
2
+
e
2
=
1
+
4
e
2
.
(3)
The
residue theorem
now gives
g
(
x
)
=
⎩
⎨
⎧
−
2
3
πi
,
2
πi
(
1
+
e
2
)
,
0
<
x
<
2
,
x
>
2.
(4)
Ancestors
(10)
3B
Paper 1
ib
2021
Past exam of the mathematics course of the University of Cambridge
Mathematics course of the University of Cambridge
Course of the University of Cambridge
University of Cambridge
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