Consider the functions f(x)=x g(x) and g(x)= [ 1 x ] Where [ ] is the greatest integer function. What is _ 1 3 ^1 f(x) d x equal to ?

Step-by-step solution

aligned & _ 1 / 3 ^1 f(x) d x= _ 1 / 3 ^ 1 / 2 f(x) d x+ _ 1 / 2 ^1 f(x) d x...(1) & f ( x )= xg ( x ) ...(1) aligned g ( 1 3 )=3 g ( 1 2 )=2 g(1)=1 The value of g(x) in value ( 1 2 , 1 3 ) will be 2 and in range ( 1 2 , 1 ) it will be 1 from (1) aligned & _ 1 / 3 ^1 f(x) d x= _ 1 / 3 ^ 1 / 2 x g(x) d x+ _ 1 / 2 ^1 x g(x) d x & = _ 1 / 3 ^ 1 / 2 x 2 d x+ _ 1 / 2 ^1 x 1 d x & = [x^2 ]_ 1 / 3 ^ 1 / 2 + [ x^2 2 ]_ 1 / 2 ^1 & = [ 1 4 - 1 9 ]+ 1 2 [1- 1 4 ] aligned = 5 36 + 3 8 = 37 72