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sin-2x-1-cos-2-x-dx-




Question Number 112481 by bemath last updated on 08/Sep/20
 ∫ ((sin 2x)/( (√(1+cos^2 x)))) dx
$$\:\int\:\frac{\mathrm{sin}\:\mathrm{2x}}{\:\sqrt{\mathrm{1}+\mathrm{cos}\:^{\mathrm{2}} \mathrm{x}}}\:\mathrm{dx}\: \\ $$
Answered by john santu last updated on 08/Sep/20
  ∫ ((2sin x cos x)/( (√(1+cos^2 x)))) dx = ∫− ((2u du)/( (√(1+u^2 ))))  where u = cos x  =−∫ ((d(1+u^2 ))/( (√(1+u^2 )))) = −2(√(1+u^2 )) + c  = −2(√(1+cos^2 x)) + c
$$\:\:\int\:\frac{\mathrm{2sin}\:{x}\:\mathrm{cos}\:{x}}{\:\sqrt{\mathrm{1}+\mathrm{cos}\:^{\mathrm{2}} {x}}}\:{dx}\:=\:\int−\:\frac{\mathrm{2}{u}\:{du}}{\:\sqrt{\mathrm{1}+{u}^{\mathrm{2}} }} \\ $$$${where}\:{u}\:=\:\mathrm{cos}\:{x} \\ $$$$=−\int\:\frac{{d}\left(\mathrm{1}+{u}^{\mathrm{2}} \right)}{\:\sqrt{\mathrm{1}+{u}^{\mathrm{2}} }}\:=\:−\mathrm{2}\sqrt{\mathrm{1}+{u}^{\mathrm{2}} }\:+\:{c} \\ $$$$=\:−\mathrm{2}\sqrt{\mathrm{1}+\mathrm{cos}\:^{\mathrm{2}} {x}}\:+\:{c}\: \\ $$

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