∫0^1000 e^|x] d x is equal to {x}= fractional part of x (a) 1000 e (b) 999(e-1) (c) 1000(e-1) (d

step 1 In this question we have to find the value of integration and we have given that factional part

∫0^1000 e^|x] d x is equal to {x}= fractional part of x (a)...

$\int_{0}^{1000} \mathrm{e}^{\mid \mathrm{x}]} \mathrm{d} \mathrm{x}$ is equal to $\{\mathrm{x}\}=$ fractional part of $\mathrm{x}$
(a) $1000 \mathrm{e}$
(b) $999(\mathrm{e}-1)$
(c) $1000(\mathrm{e}-1)$
(d) $1000(\mathrm{e}+1)$

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Key Concepts

Fractional Part Function

The fractional part function, typically denoted by {x}, gives the non?integer part of a real number, defined as {x} = x - floor(x). This function is periodic with period 1, meaning its behavior repeats on every interval of length 1, which is especially useful when integrating functions that depend on the fractional part of the variable.

Periodicity and Subinterval Decomposition

When dealing with integrals where the integrand involves a periodic function like the fractional part function, one can leverage the periodicity to split the integral into a sum of integrals over one period. This decomposition simplifies the computation by reducing a potentially large interval to repeated evaluations over the basic period.

Change of Variables in Integration

A common technique in integration involves making a substitution to take advantage of the structure of the integrand. In problems involving the fractional part function, changing the variable to isolate the fractional part (for example, writing x = k + t where t is the fractional component) can transform the integral into a simpler form that is easier to evaluate.

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