3.4b Laws of Definite Integrals

3.4b Laws of Definite Integrals




Example:
Given that 3 7 f ( x ) d x = 5 , find the values for each of the following:

(a) 3 7 6 f ( x ) d x (b) 3 7 [ 3 f ( x ) ] d x (c) 7 3 2 f ( x ) d x (d) 3 4 f ( x ) d x + 4 5 f ( x ) d x + 3 7 f ( x ) d x (e) 3 7 f ( x ) + 7 2 d x


Solution:
(a) 3 7 6 f ( x ) d x = 6 3 7 f ( x ) d x = 6 ( 5 ) = 30 (b) 3 7 [ 3 f ( x ) ] d x = 3 7 3 d x 3 7 f ( x ) d x = [ 3 x ] 3 7 5 = [ 3 ( 7 ) 3 ( 3 ) ] 5 = 7 (c) 7 3 2 f ( x ) d x = 3 7 2 f ( x ) d x = 2 3 7 f ( x ) d x = 2 ( 5 ) = 10 (d) 3 4 f ( x ) d x + 4 5 f ( x ) d x + 3 7 f ( x ) d x = 3 7 f ( x ) d x = 5 (e) 3 7 f ( x ) + 7 2 d x = 3 7 [ 1 2 f ( x ) + 7 2 ] d x = 3 7 1 2 f ( x ) d x + 3 7 7 2 d x = 1 2 3 7 f ( x ) d x + [ 7 x 2 ] 3 7 = 1 2 ( 5 ) + [ 7 ( 7 ) 2 7 ( 3 ) 2 ] = 5 2 + 14 = 16 1 2


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