0-100-sin-x-x-pi-dx-

Question Number 83225 by 09658867628 last updated on 28/Feb/20

\underset{0}{\int^{100}} \sin (x - [x]) π d x =

Commented by mathmax by abdo last updated on 29/Feb/20

A = \sum_{n = 0}^{99} \int_{n}^{n + 1} s i n ((x - n) π) d x

= \sum_{n = 0}^{99} \int_{n}^{n + 1} s i n (π x - n π) d x = \sum_{n = 0}^{99} \int_{n}^{n + 1} {(- 1)}^{n} s i n (π x) d x

= \sum_{n = 0}^{99} {(- 1)}^{n} {[- \frac{1}{π} c o s (π x)]}_{n}^{n + 1}

= \frac{1}{π} \sum_{n = 0}^{99} {(- 1)}^{n + 1} {c o s (n + 1) π - c o s (n π)}

= \frac{1}{π} \sum_{n = 0}^{99} {(- 1)}^{n + 1} {{(- 1)}^{n + 1} - {(- 1)}^{n}}

= - \frac{2}{π} \sum_{n = 0}^{99} {(- 1)}^{n} \times {(- 1)}^{n + 1} = \frac{2}{π} \sum_{n = 0}^{99} (1)

= \frac{200}{π}