Analysis and Approaches for IBDP Maths Ebook 2 | Page 276

Your Practice Set – Analysis and Approaches for IBDP Mathematics
Exercise 81
1. You are asked to investigate the integration by reduction formulae for two specific
integrals.
Consider
( ) x
I n �
1�
x
d
n
1
� x and
0 2
( ) x
H n �
( x �1) 1�x
d
�
n
1
x , 0,1, 2,
0 2 2
n � .
(a) (1) Find I (0) .
(2) Find I (1) .
(3) Using integration by parts, show that
n � 2, 3, 4, .
n �1
I( n) � I( n � 2) for
n
(4) Hence, find I (3) .
�
cos�
4
(b) (1) Using the substitution x � tan�
, show that H(0) � � d�
.
0
2
1�
2sin �
[12]
(2) Hence, show that
2
H(0) � I(0)
.
2
(c) (1) Express H (2) in terms of I (0) and H (0) .
[7]
(2) Express H (3) in terms of I (1) and H (1) .
(3) Hence, suggest a formula for Hn ( ) in terms of In� ( 2) and Hn� ( 2) for
(d) Find H (4) .
n � 2, 3, 4, .
[5]
[4]
266
SE Production Limited