prove-that-0-2020-x-x-x-x-dx-808-

Question Number 88177 by M±th+et£s last updated on 08/Apr/20

p r o v e t h a t

\int_{0}^{2020} (x - [x]) \sqrt{x - [x]} d x = 808

Answered by TANMAY PANACEA. last updated on 08/Apr/20

\int_{0}^{1} (x - 0) \sqrt{x - 0} d x + \int_{1}^{2} (x - 1) (\sqrt{x - 1} d x + . . + \int_{2019}^{2020} (x - 2019) \sqrt{x - 2019} d x

=∣ \frac{x^{\frac{5}{2}}}{\frac{5}{2}} ∣_{0}^{1} + ∣ \frac{{(x - 1)}^{\frac{5}{2}}}{\frac{5}{2}} ∣_{1}^{2} + . . + ∣ \frac{{(x - 2019)}^{\frac{5}{2}}}{\frac{5}{2}} ∣_{2019}^{2020}

\frac{5 I}{2} = (1^{\frac{5}{2}} - 0^{\frac{5}{2}}) + (1^{\frac{5}{2}} - 0^{\frac{5}{2}}) + . . + (1^{\frac{5}{2}} - 0^{\frac{5}{2}})

= 2020

I = 2 \times 404 = 808

Commented by M±th+et£s last updated on 08/Apr/20

g o d b l e s s y o u s i r

Commented by TANMAY PANACEA. last updated on 08/Apr/20

m o s t w e l c o m e s i r \dots