Analysis and Approaches for IBDP Maths Ebook 2 | Page 173

52
Example
Paper 1 Section A – Integration by Parts
12
Find � xcos�
xdx
.
[5]
Solution
Let � � sin�x
. (M1) for valid approach
d�
1
� � cos� x � d� � cos�
xdx
dx
�
1
��xcos�xdx ��
xd(sin �x)
�
1 1
xcos� xd x � ( x)(sin � x) �
�
�
sin�
xdx
�
1 1 � 1 �
xcos� xdx � xsin� x � �� cos�
x��
C
� � � � �
A1
1 1
� xcos� xdx xsin� x cos�
x C
2
�
�
A1
� � (A2) for correct working
[5]
Exercise 52
1. Find
4
ln x
� d x .
1
4
[5]
2. It is given that
4
15
� x ln xdx � k ln 4 � . Find the value of k .
1
4
[6]
3. Find
� x 2 sin 4xdx
.
[6]
4. Show that
3
10
�
3x
3�
� e cos xd x � � ( e �1)
.
0
[7]
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