Question: $\int _ { 0 } ^ { \infty }$[2 e<sup>-x</sup>] dx, where [.] denotes the greatest integer function,...

Question

$\int _ { 0 } ^ { \infty }$ [2 e^-x] dx, where [.] denotes the greatest integer function, is equal to

Answer 1

Since 0 < 2 e^-x ≤ 2 ∀ x ∈ [0, ∞), [2 e^-x] = 0, for x > ln2.

Hence the given integral = $\int _ { 0 } ^ { \ln 2 } d x$ + 0 = ln 2

Question: \(\int _ { 0 } ^ { \infty }\)[2 e<sup>-x</sup>] dx, where [.] denotes the greatest integer function,...