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Question Number 77347 by key of knowledge last updated on 05/Jan/20

if ∫sin(f(x))dx=g(x)  ∫cos(f(x))dx=?

$$\mathrm{if}\:\int\mathrm{sin}\left(\mathrm{f}\left(\mathrm{x}\right)\right)\mathrm{dx}=\mathrm{g}\left(\mathrm{x}\right) \\ $$$$\int\mathrm{cos}\left(\mathrm{f}\left(\mathrm{x}\right)\right)\mathrm{dx}=? \\ $$

Commented by mind is power last updated on 05/Jan/20

non close forme answer depend in functio f

$$\mathrm{non}\:\mathrm{close}\:\mathrm{forme}\:\mathrm{answer}\:\mathrm{depend}\:\mathrm{in}\:\mathrm{functio}\:\mathrm{f}\: \\ $$$$ \\ $$

Answered by Henri Boucatchou last updated on 06/Jan/20

∫cos(f(x))dx=∫g′′(x)dx=g′(x)+Cte

$$\int\boldsymbol{{cos}}\left(\boldsymbol{{f}}\left(\boldsymbol{{x}}\right)\right)\boldsymbol{{dx}}=\int{g}''\left({x}\right){dx}=\boldsymbol{{g}}'\left(\boldsymbol{{x}}\right)+\boldsymbol{{Cte}} \\ $$

Commented by mr W last updated on 06/Jan/20

it′s wrong sir.  sin (f(x))=g′(x)  cos (f(x))×f ′(x)=g′′(x)  ⇒cos (f(x))=((g′′(x))/(f ′(x)))≠g′(x)

$${it}'{s}\:{wrong}\:{sir}. \\ $$$$\mathrm{sin}\:\left({f}\left({x}\right)\right)={g}'\left({x}\right) \\ $$$$\mathrm{cos}\:\left({f}\left({x}\right)\right)×{f}\:'\left({x}\right)={g}''\left({x}\right) \\ $$$$\Rightarrow\mathrm{cos}\:\left({f}\left({x}\right)\right)=\frac{{g}''\left({x}\right)}{{f}\:'\left({x}\right)}\neq{g}'\left({x}\right) \\ $$

Answered by john santu last updated on 06/Jan/20

sin (f(x))= g′(x)   cos (f(x))=(√(1−sin^2 (f(x))))  cos (f(x)) =(√(1−(g′(x))^2 ))  let g′(x) = u ⇒∫cos (f(x))dx  = ∫(√(1−u^2 )) ((du/(u′)))

$$\mathrm{sin}\:\left(\mathrm{f}\left(\mathrm{x}\right)\right)=\:\mathrm{g}'\left(\mathrm{x}\right)\: \\ $$$$\mathrm{cos}\:\left(\mathrm{f}\left(\mathrm{x}\right)\right)=\sqrt{\mathrm{1}−\mathrm{sin}\:^{\mathrm{2}} \left(\mathrm{f}\left(\mathrm{x}\right)\right)} \\ $$$$\mathrm{cos}\:\left(\mathrm{f}\left(\mathrm{x}\right)\right)\:=\sqrt{\mathrm{1}−\left(\mathrm{g}'\left(\mathrm{x}\right)\right)^{\mathrm{2}} } \\ $$$$\mathrm{let}\:\mathrm{g}'\left(\mathrm{x}\right)\:=\:\mathrm{u}\:\Rightarrow\int\mathrm{cos}\:\left(\mathrm{f}\left(\mathrm{x}\right)\right)\mathrm{dx} \\ $$$$=\:\int\sqrt{\mathrm{1}−\mathrm{u}^{\mathrm{2}} }\:\left(\frac{\mathrm{du}}{\mathrm{u}'}\right) \\ $$

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