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Question Number 116501 by bemath last updated on 04/Oct/20

∫ ((sin x−cos x)/( (√(sin 2x)))) dx ?

$$\int\:\frac{\mathrm{sin}\:\mathrm{x}−\mathrm{cos}\:\mathrm{x}}{\:\sqrt{\mathrm{sin}\:\mathrm{2x}}}\:\mathrm{dx}\:? \\ $$

Answered by TANMAY PANACEA last updated on 04/Oct/20

(d/dx)(sinx+cosx)=cosx−sinx  ∫((−d(sinx+cosx))/( (√(1−1+sin2x))))  ∫((−d(sinx+cosx))/( (√((sinx+cosx)^2 −1))))  ∫(dt/( (√(t^2 −1)))) formula

$$\frac{{d}}{{dx}}\left({sinx}+{cosx}\right)={cosx}−{sinx} \\ $$$$\int\frac{−{d}\left({sinx}+{cosx}\right)}{\:\sqrt{\mathrm{1}−\mathrm{1}+{sin}\mathrm{2}{x}}} \\ $$$$\int\frac{−{d}\left({sinx}+{cosx}\right)}{\:\sqrt{\left({sinx}+{cosx}\right)^{\mathrm{2}} −\mathrm{1}}} \\ $$$$\int\frac{{dt}}{\:\sqrt{{t}^{\mathrm{2}} −\mathrm{1}}}\:{formula} \\ $$

Commented by bemath last updated on 04/Oct/20

= −ln ∣t+(√(t^2 −1)) ∣ + c   = −ln ∣sin x+cos x+(√(sin 2x)) ∣ + c ?

$$=\:−\mathrm{ln}\:\mid\mathrm{t}+\sqrt{\mathrm{t}^{\mathrm{2}} −\mathrm{1}}\:\mid\:+\:\mathrm{c}\: \\ $$$$=\:−\mathrm{ln}\:\mid\mathrm{sin}\:\mathrm{x}+\mathrm{cos}\:\mathrm{x}+\sqrt{\mathrm{sin}\:\mathrm{2x}}\:\mid\:+\:\mathrm{c}\:? \\ $$

Commented by MJS_new last updated on 04/Oct/20

yes

$$\mathrm{yes} \\ $$

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