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Question Number 39336 by rahul 19 last updated on 05/Jul/18

Commented by ajfour last updated on 05/Jul/18

$$-2isevenorodd?$$

Commented by rahul 19 last updated on 05/Jul/18

even.

Commented by ajfour last updated on 05/Jul/18

$$makesure.$$

Commented by rahul 19 last updated on 06/Jul/18

$$\mathrm{pls}\mathrm{help}....$$$$\mathrm{Mrw}3,\mathrm{Ajfour},\mathrm{Tanmay}\mathrm{sirs}....$$

Commented by abdo mathsup 649 cc last updated on 08/Jul/18

$$wehavef\left(x\right)=\mid x-\left[x\right]\mid if\left[x\right]isoddand$$$$f\left(x\right)=x-[x+1]if\left[x\right]even$$$${\int }_{-2}^{4}f\left(x\right)dx=\sum _{k=-2}^3{\int }_k^{k+1}f\left(x\right)dx$$$$={\int }_{-2}^{-1}(x-(-2)-1)dx+{\int }_{-1}^0\mid x-(-1)\mid dx$$$$+{\int }_0^1(x-0-1)dx+{\int }_1^2\mid x-1\mid dx+{\int }_2^3(x-2-1)dx$$$$+{\int }_3^4\mid x-3\mid dx$$$$={\int }_{-2}^{-1}(x+1)dx+{\int }_{-1}^o(x+1)dx+{\int }_0^1(x-1)dx$$$$+{\int }_1^2(x-1)dx+{\int }_2^3(x-3)dx+{\int }_3^4(x-3)dx$$$$={\int }_{-2}^o(x+1)dx+{\int }_0^2(x-1)dx+{\int }_2^4(x-3)dx$$$$={[\frac{x^2}{2}+x]}_{-2}^0+{[\frac{x^2}{2}-x]}_0^2+{[\frac{x^2}{2}-3x]}_2^4$$$$=-2+2+0+(8-12)-(2-6)$$$$=-4+4=0$$$${\int }_{-2}^{4}f\left(x\right)dx=0.$$$$$$

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