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Question Number 130255 by stelor last updated on 23/Jan/21

$$\mathrm{please}....\mathrm{I}\mathrm{am}\mathrm{confiouse}....1\mathrm{and}2.$$$$1.\int \mid x\mid \mathrm{d}x=??$$$$2.\mathrm{with}{\left(\mathrm{F}\left(x\right)\right)}^{{}^{\prime }}=\mathrm{f}\left(x\right)\int \mid \mathrm{f}\left(x\right)\mid dx=??$$$$$$

Answered by Olaf last updated on 23/Jan/21

$$\int \mid x\mid dx=\{\begin{array}{l}-\frac{1}{2}{x}^2+\mathrm{C},x⩽0\\ +\frac{1}{2}{x}^2+\mathrm{C},x⩾0\end{array}$$$$\Rightarrow \int \mid x\mid dx=\frac{1}{2}x\mid x\mid +\mathrm{C}$$$$$$$$\int \mid f\left(x\right)\mid dx=\{\begin{array}{l}-\mathrm{F}\left(x\right)+\mathrm{C},f\left(x\right)⩽0\\ +\mathrm{F}\left(x\right)+\mathrm{C},f\left(x\right)⩾0\end{array}$$

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