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Question Number 6390 by FilupSmith last updated on 25/Jun/16

$$\mathrm{Solve}:$$$$I\left(n\right)={\int }_0^n{(-1)}^{\lfloor x\rfloor }{x}^{2}dx$$

Commented by prakash jain last updated on 25/Jun/16

$$I\left(n\right)={\int }_0^1{x}^{2}dx-{\int }_1^2{x}^{2}dx+-...+{(-1)}^{n-1}{\int }_{n-1}^n{x}^{2}dx$$$$=\frac{1^3}{3}-(\frac{2^3}{3}-\frac{1^3}{3})+(\frac{3^3}{3}-\frac{2^3}{3})+-..+{(-1)}^{n-1}(\frac{n^3}{3}-\frac{{(n-1)}^3}{3})$$

Commented by FilupSmith last updated on 26/Jun/16

$$\mathrm{I}\mathrm{love}\mathrm{this}\mathrm{approach}!\mathrm{i}\mathrm{didnt}\mathrm{even}\mathrm{think}$$$$\mathrm{of}\mathrm{it}\mathrm{this}\mathrm{way}!$$

Commented by malwaan last updated on 26/Jun/16

$$thisisrightfor{(-1)}^{n}not{(-1)}^{\mid x\mid }$$

Commented by prakash jain last updated on 26/Jun/16

$$\lfloor x\rfloor$$$$0for0⩽x<1$$$$1for1⩽x<2$$$$andsoon$$

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