let-f-defined-on-0-1-by-f-0-0-and-f-x-1-2-1-2x-1-calculate-0-1-f-x-dx-

Question Number 51995 by maxmathsup by imad last updated on 01/Jan/19

l e t f d e f i n e d o n [0, 1] b y f (0) = 0 a n d f (x) = \frac{1}{2 [\frac{1}{2 x}] + 1}

c a l c u l a t e \int_{0}^{1} f (x) d x

Answered by tanmay.chaudhury50@gmail.com last updated on 03/Jan/19

x \frac{1}{x} \frac{1}{2 x} [\frac{1}{2 x}] 2 [\frac{1}{2 x}] + 1 \frac{1}{2 [\frac{1}{2 x}] + 1}

1.0 1 0.5 0 1 1

0.9 1.11 0.55 0 1 1

\dots . .

0.51 1.96 0.98 0 1 1

0.5 2 1 1 3 \frac{1}{3}

0.4 2.5 1.25 1 3 \frac{1}{3}

0.3 3.33 1.67 1 3 \frac{1}{3}

0.2 5 2.5 2 5 \frac{1}{5}

\int_{0}^{1} \frac{1}{2 [\frac{1}{2 x}] + 1} d x

+ \int_{0.2}^{0.3} \frac{1}{5} + \int_{0.3}^{0.5} \frac{1}{3} d x + \int_{0.5}^{1} 1 d x

w a i t \dots