AP EAMCET · Maths · Definite Integration
If [.] denotes the greatest integer function, then \(\int_0^{1000} e^{x-[x]} d x=\)
- A \(\frac{\mathrm{e}^{1000}-1}{1000}\)
- B \(1000(\mathrm{e}-1)\)
- C \(\frac{\mathrm{e}^{1000}-1}{\mathrm{e}-1}\)
- D \(\frac{\mathrm{e}-1}{1000}\)
Answer & Solution
Correct Answer
(B) \(1000(\mathrm{e}-1)\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & \text {} \int_0^{1000} \mathrm{e}^{\mathrm{x}-[\mathrm{x}]} \mathrm{dx} \\ & =\int_0^1 \mathrm{e}^{\mathrm{x}-0} \mathrm{dx}+\int_1^2 \mathrm{e}^{\mathrm{x}-1} \mathrm{dx}+\int_2^3 \mathrm{e}^{\mathrm{x}-2} \mathrm{dx}+\ldots+\int_{999}^{1000}…
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