calculate-the-value-of-0-1000-e-x-x-dx-where-x-is-greatest-integer-functon-

Question Number 22843 by gourav~ last updated on 22/Oct/17

c a l c u l a t e t h e v a l u e o f \int_{0}^{1000} e^{x - [x]} d x = ?

w h e r e \dots . [x] i s g r e a t e s t i n t e g e r f u n c t o n

Answered by ajfour last updated on 23/Oct/17

l e t I = \int_{0}^{1000} e^{x - [x]} d x

= \underset{0}{\sum^{999}} \int_{m}^{m + 1} e^{x - [x]} d x

l e t x - [x] = {x}

\Rightarrow d x = d {x} f o r e a c h i n t e r v a l

s o I = \underset{0}{\sum^{999}} \int_{0}^{1} e^{{x}} d {x} = 1000 (e - 1) .

Commented by gourav~ last updated on 23/Oct/17

y e s s i r \dots b u t h o w

Commented by jota+ last updated on 03/Dec/17

m ⩽ x < m + 1 \Rightarrow x - [x] = x - m = u

t h e n x = m + u

\int_{m}^{m + 1} d x \to \int_{0}^{1} d u

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