Question Number 45495 by Meritguide1234 last updated on 13/Oct/18
Answered by tanmay.chaudhury50@gmail.com last updated on 13/Oct/18
Commented by tanmay.chaudhury50@gmail.com last updated on 14/Oct/18
$$\int_{{a}−\mathrm{2}{b}} ^{\mathrm{2}{a}−{b}} \mid\sqrt{\left({x}+\mathrm{3}{b}\right)\left(\left(\mathrm{2}{a}−{b}−{x}\right)\right.}\:\:−\sqrt{\left(\mathrm{3}{a}−{x}\right)\left(\mathrm{2}{b}−{a}+{x}\right)}\:\mid{dx} \\ $$$$\int_{{a}−\mathrm{2}{b}} ^{\mathrm{2}{a}−{b}} \sqrt{\left(\mathrm{3}{a}−\mathrm{3}{b}−{x}+\mathrm{3}{b}\right)×\left(\mathrm{2}{a}−{b}−\left(\mathrm{3}{a}−\mathrm{3}{b}−{x}\right)\right.}\:\:\:−\sqrt{\left(\mathrm{3}{a}−{x}\right)\left(\mathrm{2}{b}−{a}+{x}\right)}\:\mid{dx} \\ $$$$\int_{{a}−\mathrm{2}{b}} ^{\mathrm{2}{a}−{b}} \mid\sqrt{\left(\mathrm{3}{a}−{x}\right)\left(\mathrm{2}{b}−{a}−{x}\right)}\:−\sqrt{\left({x}+\mathrm{3}{b}\right)\left(\mathrm{2}{a}−{b}−{x}\right)}\:{dx} \\ $$$$\int_{{a}−\mathrm{2}{b}} ^{\mathrm{2}{a}−{b}} \mid{p}−{q}\mid{dx}=\int_{{a}−\mathrm{2}{b}} ^{\mathrm{2}{a}−{b}} \mid{q}−{p}\mid{dx} \\ $$$${that}\:{means}\:{p}−{q}={q}−{p} \\ $$$$\mathrm{2}{p}=\mathrm{2}{q}\:\:\:\:{p}={q} \\ $$$$\left({x}+\mathrm{3}{b}\right)\left(\mathrm{2}{a}−{b}−{x}\right)=\left(\mathrm{3}{a}−{x}\right)\left(\mathrm{2}{b}−{a}−{x}\right) \\ $$$$\mathrm{2}{ax}−{bx}−{x}^{\mathrm{2}} +\mathrm{6}{ab}−\mathrm{3}{b}^{\mathrm{2}} −\mathrm{3}{bx}=\mathrm{6}{ab}−\mathrm{3}{a}^{\mathrm{2}} −\mathrm{3}{ax}−\mathrm{2}{bx}+{ax}+{x}^{\mathrm{2}} \\ $$$$\mathrm{2}{ax}+\mathrm{2}{ax}−\mathrm{4}{bx}+\mathrm{2}{bx}−{x}^{\mathrm{2}} −{x}^{\mathrm{2}} +\mathrm{6}{ab}−\mathrm{6}{ab}=\mathrm{0} \\ $$$$\mathrm{4}{ax}−\mathrm{2}{bx}−\mathrm{2}{x}^{\mathrm{2}} =\mathrm{0} \\ $$$$\mathrm{2}{x}\left(\mathrm{2}{a}−{b}−{x}\right)=\mathrm{0} \\ $$$${x}=\mathrm{2}{a}−{b} \\ $$$$ \\ $$$$ \\ $$