find-the-integration-ln-x-dx-x-gt-2-

Question Number 89867 by M±th+et£s last updated on 19/Apr/20

f i n d t h e i n t e g r a t i o n

\int l n ⌊ x ⌋ d x; x > 2

Commented by ~blr237~ last updated on 20/Apr/20

Answered by mr W last updated on 20/Apr/20

l e t x = n + f

d x = d f

\int \ln ⌊ x ⌋ d x = \ln n \int d f = (\ln n) f + C

= (\ln ⌊ x ⌋) (x - ⌊ x ⌋) + C

Commented by M±th+et£s last updated on 20/Apr/20

s i r w h y \int l n ⌊ x ⌋ d x = l n (n) \int d f a n d h o w

d i d y o u g e t l n (⌊ x ⌋) (x - ⌊ x ⌋)

Commented by mr W last updated on 21/Apr/20

x = n + f w i t h n = ⌊ x ⌋, f = x - ⌊ x ⌋

i n x = n + f, n i s c o n s t a n t, t h e r e f o r e

d x = d f

\int \ln ⌊ x ⌋ d x = \int (\ln n) d f = \ln (n) \int d f

= \ln (n) f + C

= (\ln ⌊ x ⌋) (x - ⌊ x ⌋) + C

Commented by M±th+et£s last updated on 21/Apr/20

t h a n k y o u f o r e x p l a i n g t h a t f o r m e

Answered by M±th+et£s last updated on 22/Apr/20

i t r y a l o t a n d i g e t t h a t

l n [x] d x = l n (2) x \in (2, 3)

f (x) = l n [x] x \in (n, n + 1)

\int l n [x] d x = \underset{k = 2}{\sum^{n - 1}} l n (x) + \int_{n = [x]}^{x} l n [x] d x

\underset{k = 2}{\sum^{n - 1}} l n (k) + l n (n) (x - [x]) + c