2 . You are asked to investigate the integration by reduction formulae for a specific integral .
Let
� n
I( n) � � xsin xdx , where n � 0,1 , 2 , .
0
( a ) ( i ) Show that I( 0 )
1 2
2
� � .
( ii ) Find I ( 1 ) .
( b ) ( i ) By using sin x�1� cos
2 2 x, show that
� n 2
( � 2 ) � ( ) �� sin cos d 0
I n I n x x x x
[ 7 ]
� x 2 sin n x cos x d x
0
� can be expressed as n�1
1 � d ( sin x) cos �
1
0 d
x x dx n� � . x
( ii ) Express
� x 2 sin n x cos x d x
0
� in terms of In� ( 2 ) and n .
n �1 ( iii ) Hence , show that I( n �2 ) � I( n)
. n � 2
( c ) By using ( b )( iii ), find , in terms of � ,
[ 11 ]
( i ) I ( 4 ) ;
( ii ) I ( 7 ) .
( d ) Explain why I( 2 n) � I( 2n �1 ) � I( 2n � 2 ) for n � 1.
[ 6 ]
[ 2 ]
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