Question

The question is in the attachment!!​

Answer 1

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`\qquad\quad\hookrightarrow{\sf { 1000(e-1)}}`

Solution:

`\begin{aligned}&\longrightarrow \int_(0) ^(1000) {e}^( - |x| )dx = \sum_(k = 0)^(999) \int_(k)^(k + 1)e ^(x - |x| )dx \\ \\ &\longrightarrow\int_(0) ^(1000) {e}^( - |x| )dx = \sum_(k = 0)^(999) \int_(k)^(k + 1)e^(x - k)dx \\\\&\longrightarrow\int_(0) ^(1000) {e}^( - |x| )dx = \sum_(k = 0)^(999)e^{x - k \big |_(k)^ {k + 1}} \\ \\& \longrightarrow\int_(0) ^(1000) {e}^( - |x| )dx = \sum_(k = 0)^ {999}(e - 1)\\\\&\longrightarrow\int_(0) ^(1000) {e}^( - |x| )dx = 1000(e - 1)\end{aligned}`