 Compute, please, definite integral which have a absolute value function ushwaui 2021-11-19 Answered
Compute, please, definite integral which have a absolute value function like these below:
$$\displaystyle{\int_{{-{2}}}^{{{3}}}}{\left|{x}\right|}{\left.{d}{x}\right.}$$
$$\displaystyle{\int_{{-{2}}}^{{{3}}}}{\left|{x}-{1}\right|}{\left.{d}{x}\right.}$$
$$\displaystyle{\int_{{-{2}\pi}}^{{{2}\pi}}}{\left|{\sin{{x}}}\right|}{\left.{d}{x}\right.}$$

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You should consider the fact that |x| is one linear function for $$\displaystyle{x}\in{\left(-\infty;{0}\right]}$$ and a linear function for $$x\in[0;\infty)$$ to break the interval of integration into subintervals that are easier.
$$\displaystyle{\int_{{-{2}}}^{{{3}}}}{\left|{x}\right|}{\left.{d}{x}\right.}={\int_{{-{2}}}^{{{0}}}}-{x}{\left.{d}{x}\right.}+{\int_{{{0}}}^{{{3}}}}{x}{\left.{d}{x}\right.}$$

Not exactly what you’re looking for? Crom1970
Firstly, you can split each integral into multiple, smaller integrals for your first integral, such as:
$$\displaystyle{\int_{{-{2}}}^{{{0}}}}-{x}{\left.{d}{x}\right.}$$ and $$\displaystyle{\int_{{{0}}}^{{{3}}}}{x}{\left.{d}{x}\right.}$$
You can also look at the graphs and calculate the area underneath them using simple geometry.