2 . You are asked to investigate the integration by reduction formulae for a specific integral .
Consider
1 x n
I( n) � � f ( x) dx , where f ( x) �e ( 1 � x)
, n � 0,1 , 2 , .
0
( a ) Show that f( x ) is decreasing in 0�x � 1 for n � 0 .
( b ) Explain why 0 �In ( ) � 1 for n � 0 .
( c ) ( i ) Show that I( 0 ) �e� 1.
[ 3 ]
[ 2 ]
Find
( ii ) I ( 1 ) ;
( iii ) I ( 2 ) .
( d ) Show that I( n) � nI( n �1 ) � 1 for n � 0 .
( e ) Hence , show that
( f ) By using the fact that n
� 1 � I( n) � n! �e
�1�� r�1 r!
�. � �
In ( ) lim � 0 , write down n�� n! n
1 lim� . n�� � r! r
0
[ 12 ]
[ 5 ]
[ 5 ]
[ 1 ]
© SE Production Limited 8 All Rights Reserved 2021