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




Question Number 129274 by bramlexs22 last updated on 14/Jan/21
 ∫ ((cos^2 x+cos x)/(1+sin x+cos x)) dx ?
$$\:\int\:\frac{\mathrm{cos}\:^{\mathrm{2}} \mathrm{x}+\mathrm{cos}\:\mathrm{x}}{\mathrm{1}+\mathrm{sin}\:\mathrm{x}+\mathrm{cos}\:\mathrm{x}}\:\mathrm{dx}\:? \\ $$
Answered by liberty last updated on 14/Jan/21
 ∫ ((cos x(cos x+1))/(sin x+2cos^2 ((x/2)))) dx = ∫ ((cos x(2cos^2 ((x/2))))/(2cos ((x/2))[ sin ((x/2))+cos ((x/2))])) dx=   ∫ ((cos ((x/2))(cos ((x/2))+sin ((x/2)))(cos ((x/2))−sin ((x/2))))/(sin ((x/2))+cos ((x/2))))dx=   ∫ cos ((x/2))(cos ((x/2))−sin ((x/2)))dx =   ∫ ((1/2)+(1/2)cos x−(1/2)sin x)dx =           (1/2)(x+sin x+cos x) + C
$$\:\int\:\frac{\mathrm{cos}\:\mathrm{x}\left(\mathrm{cos}\:\mathrm{x}+\mathrm{1}\right)}{\mathrm{sin}\:\mathrm{x}+\mathrm{2cos}\:^{\mathrm{2}} \left(\frac{\mathrm{x}}{\mathrm{2}}\right)}\:\mathrm{dx}\:=\:\int\:\frac{\mathrm{cos}\:\mathrm{x}\left(\mathrm{2cos}\:^{\mathrm{2}} \left(\frac{\mathrm{x}}{\mathrm{2}}\right)\right)}{\mathrm{2cos}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\left[\:\mathrm{sin}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)+\mathrm{cos}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\right]}\:\mathrm{dx}= \\ $$$$\:\int\:\frac{\mathrm{cos}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\left(\mathrm{cos}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)+\mathrm{sin}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\right)\left(\mathrm{cos}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)−\mathrm{sin}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\right)}{\mathrm{sin}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)+\mathrm{cos}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)}\mathrm{dx}= \\ $$$$\:\int\:\mathrm{cos}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\left(\mathrm{cos}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)−\mathrm{sin}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\right)\mathrm{dx}\:= \\ $$$$\:\int\:\left(\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{cos}\:\mathrm{x}−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{sin}\:\mathrm{x}\right)\mathrm{dx}\:= \\ $$$$\:\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{x}+\mathrm{sin}\:\mathrm{x}+\mathrm{cos}\:\mathrm{x}\right)\:+\:\mathrm{C}\: \\ $$

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