Menu Close

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




Question Number 129177 by bramlexs22 last updated on 13/Jan/21
 G = ∫ ((sin^2 x+sin x)/(1+sin x+cos x)) dx ?
$$\:\mathrm{G}\:=\:\int\:\frac{\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}+\mathrm{sin}\:\mathrm{x}}{\mathrm{1}+\mathrm{sin}\:\mathrm{x}+\mathrm{cos}\:\mathrm{x}}\:\mathrm{dx}\:? \\ $$
Answered by liberty last updated on 13/Jan/21
 G = ∫ ((sin x(sin x+1))/(sin x+2cos^2 ((x/2)))) dx   G = ∫ ((2sin ((x/2))cos ((x/2)){sin ((x/2))+cos ((x/2))}^2 )/(2cos ((x/2)){sin ((x/2))+cos ((x/2))}))dx   G = ∫2sin ((x/2)) {sin ((x/2))+cos ((x/2))} dx   G = ∫(1−cos x+sin x) dx = x−sin x−cos x + c
$$\:\mathrm{G}\:=\:\int\:\frac{\mathrm{sin}\:\mathrm{x}\left(\mathrm{sin}\:\mathrm{x}+\mathrm{1}\right)}{\mathrm{sin}\:\mathrm{x}+\mathrm{2cos}\:^{\mathrm{2}} \left(\frac{\mathrm{x}}{\mathrm{2}}\right)}\:\mathrm{dx} \\ $$$$\:\mathrm{G}\:=\:\int\:\frac{\mathrm{2sin}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\mathrm{cos}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\left\{\mathrm{sin}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)+\mathrm{cos}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\right\}^{\mathrm{2}} }{\mathrm{2cos}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\left\{\mathrm{sin}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)+\mathrm{cos}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\right\}}\mathrm{dx} \\ $$$$\:\mathrm{G}\:=\:\int\mathrm{2sin}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\:\left\{\mathrm{sin}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)+\mathrm{cos}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\right\}\:\mathrm{dx} \\ $$$$\:\mathrm{G}\:=\:\int\left(\mathrm{1}−\mathrm{cos}\:\mathrm{x}+\mathrm{sin}\:\mathrm{x}\right)\:\mathrm{dx}\:=\:\underline{\mathrm{x}−\mathrm{sin}\:\mathrm{x}−\mathrm{cos}\:\mathrm{x}\:+\:\mathrm{c}}\: \\ $$$$ \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *