3.7 Integration, SPM Practice (Paper 1)


Question 13:
Given that y= x 2 2x1 , show that dy dx = 2x( x1 ) ( 2x1 ) 2 . Hence, evaluate  2 2 x( x1 ) 4 ( 2x1 ) 2  dx .

Solution:
y= x 2 2x1 dy dx = ( 2x1 )( 2x )x( 2 ) ( 2x1 ) 2     = 4 x 2 2x2 x 2 ( 2x1 ) 2     = 2 x 2 2x ( 2x1 ) 2     = 2x( x1 ) ( 2x1 ) 2  ( shown ) 2 2 2x( x1 ) ( 2x1 ) 2  dx = [ x 2 2x1 ] 2 2 1 8 2 2 2x( x1 ) ( 2x1 ) 2  dx = 1 8 [ x 2 2x1 ] 2 2 1 4 2 2 x( x1 ) ( 2x1 ) 2  dx = 1 8 [ ( 2 2 2( 2 )1 )( ( 2 ) 2 2( 2 )1 ) ]                            = 1 8 [ ( 4 3 )( 4 5 ) ]                            = 1 8 ( 32 15 )                            = 4 15

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