3.4c Integration as the Inverse of Differentiation


3.4c Integration as the Inverse of Differentiation

Example:
Shows that  d dx [ 2x+5 x 2 3 ]= 2( x 2 +5x+3 ) ( x 2 3 ) 2 Hence, find the value of  0 2 ( x 2 +5x+3 ) ( x 2 3 ) 2  dx

Solution:
d dx [ 2x+5 x 2 3 ]= ( x 2 3 )( 2 )( 2x+5 )( 2x ) ( x 2 3 ) 2                   = 2 x 2 64 x 2 10x ( x 2 3 ) 2                   = 2 x 2 10x6 ( x 2 3 ) 2                   = 2( x 2 +5x+3 ) ( x 2 3 ) 2 0 2 2( x 2 +5x+3 ) ( x 2 3 ) 2  dx = [ 2x+5 x 2 3 ] 0 2 2 0 2 ( x 2 +5x+3 ) ( x 2 3 ) 2  dx = [ 2x+5 x 2 3 ] 0 2      0 2 ( x 2 +5x+3 ) ( x 2 3 ) 2  dx = 1 2 [ ( 2( 2 )+5 2 2 3 )( 2( 0 )+5 0 2 3 ) ]                                  = 1 2 [ 9( 5 3 ) ]                                  = 1 2 × 32 3                                  = 16 3                                  =5 1 3

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