Analysis and Approaches for IBDP Maths Ebook 2 | Page 177

54
Example
Paper 2 Section A – Inverse Trigonometric Substitutions
12
Use the substitution
x � tan�
to show that
3 1 3 � 2
� dx
� .
1
3
2 2
2
(1 � x )
[5]
Solution
x � tan�
dx
2 2
sec � dx
sec �d�
d� � � � M1
�1
�
x � 3 ��
� tan 3 �
3
�1
�
x �1��
� tan 1 �
4
�
3 1 1
3
2
�� dx
�
1
3 � �
�sec �d�
A1
3
2 2 4
2 2
(1 �x
) (1 �tan �)
1
3
� 2
1 sec �
3
dx
� d
3 � 3 �
2 2 4
sec �
(1 � x )
� � A1
�
3 1
3
� dx
� cos d
1
3 � � � �
2 2 4
(1 � x )
A1
�
3 1
3
� dx
�
3 �sin�
��
1
2 2
4
(1 � x )
A1
3
1
3
�
�
1
1
3
2 2
(1 � x )
� �
dx
�sin �sin
3 4
1 3 2
dx
� �
3
2 2
2 2
(1 � x )
3 1 3 � 2
� dx
�
AG
1
3
2 2
2
(1 � x )
[5]
www.seprodstore.com
167