If f(x)=[x^2]-[x]^2, where [x] denotes the greatest integer <=x then in_1^2 f(x)dx?

grabrovi0u

grabrovi0u

Answered question

2022-10-22

If f ( x ) = x 2 x 2 ,where [x] denotes the greatest integer x then 1 2 f ( x ) d x ?

Answer & Explanation

Layton Leach

Layton Leach

Beginner2022-10-23Added 15 answers

On the interval (1,2) you have [x]=1. On the interval ( 1 , 2 ) you have [ x 2 ] = 1 and on the interval ( 2 , 3 ) you have [ x 2 ] = 3 and on the interval ( 3 , 4 ) you have [ x 2 ] = 3. So you can break on the integral into:
1 2 f ( x ) d x = 1 2 f ( x ) d x + 2 3 f ( x ) d x + 3 2 f ( x ) d x = 1 2 1 1 d x + 2 3 2 1 d x + 3 2 3 1 d x

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