Integration as the Inverse of Differentiation Posted on April 22, 2020 by user 3.4c Integration as the Inverse of DifferentiationExample: 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 −6−4 x 2 −10x ( x 2 −3 ) 2 = −2 x 2 −10x−6 ( 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