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




Question Number 86091 by jagoll last updated on 27/Mar/20
∫ ((sin x−cos x)/( (√(sin 2x)))) dx
$$\int\:\frac{\mathrm{sin}\:\mathrm{x}−\mathrm{cos}\:\mathrm{x}}{\:\sqrt{\mathrm{sin}\:\mathrm{2x}}}\:\mathrm{dx} \\ $$
Commented by john santu last updated on 27/Mar/20
in other way  ∫ −((d(sin x+cos x))/( (√((sin x+cos x)^2 −1))))   −∫ (du/( (√(u^2 −1)))) = −ln∣u+(√(u^2 −1)) ∣ + c  = − ln ∣sin x+cos x+(√(sin 2x)) ∣ +c
$${in}\:{other}\:{way} \\ $$$$\int\:−\frac{{d}\left(\mathrm{sin}\:{x}+\mathrm{cos}\:{x}\right)}{\:\sqrt{\left(\mathrm{sin}\:{x}+\mathrm{cos}\:{x}\right)^{\mathrm{2}} −\mathrm{1}}}\: \\ $$$$−\int\:\frac{{du}}{\:\sqrt{{u}^{\mathrm{2}} −\mathrm{1}}}\:=\:−\mathrm{ln}\mid{u}+\sqrt{{u}^{\mathrm{2}} −\mathrm{1}}\:\mid\:+\:{c} \\ $$$$=\:−\:\mathrm{ln}\:\mid\mathrm{sin}\:{x}+\mathrm{cos}\:{x}+\sqrt{\mathrm{sin}\:\mathrm{2}{x}}\:\mid\:+{c} \\ $$
Answered by jagoll last updated on 27/Mar/20
Commented by john santu last updated on 27/Mar/20
great
$${great} \\ $$

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