Question Number 225814 by fantastic last updated on 12/Nov/25
$$\int\mid{x}\mid{dx} \\ $$
Answered by Frix last updated on 13/Nov/25
$$\mathrm{By}\:\mathrm{parts} \\ $$$${u}'=\mathrm{1}\:\rightarrow\:{u}={x} \\ $$$${v}=\mid{x}\mid\:\rightarrow\:{v}'=\mathrm{sign}\:\left({x}\right) \\ $$$$\int\mid{x}\mid{dx}={x}\mid{x}\mid−\int{x}\mathrm{sign}\:\left({x}\right)\:{dx} \\ $$$$\mathrm{But}\:{x}\mathrm{sign}\:\left({x}\right)\:=\mid{x}\mid\:\Rightarrow \\ $$$$\int\mid{x}\mid{dx}={x}\mid{x}\mid−\int\mid{x}\mid{dx} \\ $$$$\Rightarrow \\ $$$$\int\mid{x}\mid{dx}=\frac{{x}\mid{x}\mid}{\mathrm{2}}+{C} \\ $$
Commented by fantastic last updated on 13/Nov/25
$${thanks}\:{sir} \\ $$
Commented by Frix last updated on 13/Nov/25
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