Definite Integrals Posted on April 22, 2020 by user 3.4a Definite Integral of f(x) from x=a to x=b Example: Evaluate each of the following. (a) ∫ − 1 0 ( 3 x 2 − 2 x + 5 ) d x (b) ∫ 0 2 ( 2 x + 1 ) 3 d x Solution: (a) ∫ − 1 0 ( 3 x 2 − 2 x + 5 ) d x = [ 3 x 3 3 − 2 x 2 2 + 5 x ] − 1 0 = [ x 3 − x 2 + 5 x ] − 1 0 = 0 − [ ( − 1 ) 3 − ( − 1 ) 2 + 5 ( − 1 ) ] = 0 − ( − 1 − 1 − 5 ) = 7 (b) ∫ 0 2 ( 2 x + 1 ) 3 d x = [ ( 2 x + 1 ) 4 4 ( 2 ) ] 0 2 = [ ( 2 x + 1 ) 4 8 ] 0 2 = [ ( 2 ( 2 ) + 1 ) 4 8 ] − [ ( 2 ( 0 ) + 1 ) 4 8 ] = 625 8 − 1 8 = 78
