INTEGRATING ROUNDING FUNCTIONS-(II)

23 03 2009

SQUARE FRACTIONAL PART DEFINITE INTEGRAL:

For any \displaystyle x\geq 0:

\displaystyle I_1= \int_0^x \left\{{x}\right\}^2 dx = \int_0^{\lfloor x \rfloor} \left\{{x}\right\}^2 dx+ \int_{{\lfloor x \rfloor}}^ {x} \left\{{x}\right\}^2 dx

The function \displaystyle \left\{{x}\right\}^2 has a periodical behaviour, so the limits can be reduced to the first period:

\displaystyle I_1= \lfloor x \rfloor \int_0^{1} {\left\{{x}\right\}^2 dx}+ \int_0^{\left\{{x}\right\}}{ \left\{{x}\right\}^2 dx} = \lfloor x \rfloor \int_0^1 x^2 dx+ \int_0^{\left\{{x}\right\}} {x^2 dx}

\displaystyle I_1= \frac{ \lfloor x \rfloor + \left\{{x}\right\}^3}{3}

\displaystyle I_1= \int_0^x \left\{{x}\right\}^2 dx= \frac{x+ \left\{{x}\right\}^{3}-\left\{{x}\right\}}{3}

POWER FRACTIONAL PART DEFINITE INTEGRAL:

The same formula can be extended to any given positive integer power:

\displaystyle I_2= \int_0^x \left\{{x}\right\}^n dx= \frac{x+ \left\{{x}\right\}^{n+1}-\left\{{x}\right\}}{n+1}

\displaystyle I_2= \frac{ \lfloor x \rfloor + \left\{{x}\right\}^{n+1}}{n+1}

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