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B-0-2-4-x-x-x-4-x-dx-




Question Number 124135 by bramlexs22 last updated on 01/Dec/20
 B = ∫_0 ^2  [(√((4−x)/x)) − (√(x/(4−x))) ] dx
$$\:{B}\:=\:\underset{\mathrm{0}} {\overset{\mathrm{2}} {\int}}\:\left[\sqrt{\frac{\mathrm{4}−{x}}{{x}}}\:−\:\sqrt{\frac{{x}}{\mathrm{4}−{x}}}\:\right]\:{dx} \\ $$
Answered by nico last updated on 01/Dec/20
=∫_0 ^2 ((4−x−x)/( (√(4x−x^2 ))))dx=∫_0 ^2 ((d(4x−x^2 ))/( (√(4x−x^2 ))))  =2(√(4x−x^2 ))∣_0 ^2 =4
$$=\int_{\mathrm{0}} ^{\mathrm{2}} \frac{\mathrm{4}−{x}−{x}}{\:\sqrt{\mathrm{4}{x}−{x}^{\mathrm{2}} }}{dx}=\int_{\mathrm{0}} ^{\mathrm{2}} \frac{{d}\left(\mathrm{4}{x}−{x}^{\mathrm{2}} \right)}{\:\sqrt{\mathrm{4}{x}−{x}^{\mathrm{2}} }} \\ $$$$=\mathrm{2}\sqrt{\mathrm{4}{x}−{x}^{\mathrm{2}} }\mid_{\mathrm{0}} ^{\mathrm{2}} =\mathrm{4} \\ $$

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