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# if-sin-f-x-dx-g-x-cos-f-x-dx-

Question Number 77347 by key of knowledge last updated on 05/Jan/20
$$\mathrm{if}\:\int\mathrm{sin}\left(\mathrm{f}\left(\mathrm{x}\right)\right)\mathrm{dx}=\mathrm{g}\left(\mathrm{x}\right) \\$$$$\int\mathrm{cos}\left(\mathrm{f}\left(\mathrm{x}\right)\right)\mathrm{dx}=? \\$$
Commented by mind is power last updated on 05/Jan/20
$$\mathrm{non}\:\mathrm{close}\:\mathrm{forme}\:\mathrm{answer}\:\mathrm{depend}\:\mathrm{in}\:\mathrm{functio}\:\mathrm{f}\: \\$$$$\\$$
Answered by Henri Boucatchou last updated on 06/Jan/20
$$\int\boldsymbol{{cos}}\left(\boldsymbol{{f}}\left(\boldsymbol{{x}}\right)\right)\boldsymbol{{dx}}=\int{g}''\left({x}\right){dx}=\boldsymbol{{g}}'\left(\boldsymbol{{x}}\right)+\boldsymbol{{Cte}} \\$$
Commented by mr W last updated on 06/Jan/20
$${it}'{s}\:{wrong}\:{sir}. \\$$$$\mathrm{sin}\:\left({f}\left({x}\right)\right)={g}'\left({x}\right) \\$$$$\mathrm{cos}\:\left({f}\left({x}\right)\right)×{f}\:'\left({x}\right)={g}''\left({x}\right) \\$$$$\Rightarrow\mathrm{cos}\:\left({f}\left({x}\right)\right)=\frac{{g}''\left({x}\right)}{{f}\:'\left({x}\right)}\neq{g}'\left({x}\right) \\$$
Answered by john santu last updated on 06/Jan/20
$$\mathrm{sin}\:\left(\mathrm{f}\left(\mathrm{x}\right)\right)=\:\mathrm{g}'\left(\mathrm{x}\right)\: \\$$$$\mathrm{cos}\:\left(\mathrm{f}\left(\mathrm{x}\right)\right)=\sqrt{\mathrm{1}−\mathrm{sin}\:^{\mathrm{2}} \left(\mathrm{f}\left(\mathrm{x}\right)\right)} \\$$$$\mathrm{cos}\:\left(\mathrm{f}\left(\mathrm{x}\right)\right)\:=\sqrt{\mathrm{1}−\left(\mathrm{g}'\left(\mathrm{x}\right)\right)^{\mathrm{2}} } \\$$$$\mathrm{let}\:\mathrm{g}'\left(\mathrm{x}\right)\:=\:\mathrm{u}\:\Rightarrow\int\mathrm{cos}\:\left(\mathrm{f}\left(\mathrm{x}\right)\right)\mathrm{dx} \\$$$$=\:\int\sqrt{\mathrm{1}−\mathrm{u}^{\mathrm{2}} }\:\left(\frac{\mathrm{du}}{\mathrm{u}'}\right) \\$$