Analysis and Approaches for IBDP Maths Ebook 2 | Page 175

(b)
f 1 1
( x )d x �
�
� � � �d
2(1 �4 x) 2(1 �4 x)
x
�
�
� �
� 1 1 1 1
f ( x)dx ln(1 4 x) � �
� � � � ��� �ln(1 � 4 x)
� C
2 4 2 � 4 �
A1
1 1
� f ( x)dx � ln(1 � 4 x) � ln(1 � 4 x)
� C
A1
8 8
[2]
12
Exercise 53
1
1. Let f( x)
�
2
x �2x� 8
, x � 4 .
(a) Express f( x ) in partial fractions.
(b) Hence, find � f ( x)dx
.
[4]
[1]
3
2. Let f( x)
�
2
x � 5x
, x � 5.
(a) Express f( x ) in partial fractions.
(b)
Hence, find
b� .
11
� f ( t )dt
, giving the answer in the form aln11� bln6
, where a ,
6
[4]
[2]
x � 4
3. Let f( x)
�
2
x �3x� 2
, x �� 2 , x �� 1.
(a) Write down the equations of the vertical asymptotes of f( x ).
(b) Express f( x ) in partial fractions.
(c)
Hence, find
5
� f ( x )dx
.
4
[2]
[4]
[2]
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165