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Question Number 22623 by tawa tawa last updated on 21/Oct/17
How is:   ∫ ((x + sin(x))/(1 + cos(x))) dx = xtan((x/2)) + C
$$\mathrm{How}\:\mathrm{is}:\:\:\:\int\:\frac{\mathrm{x}\:+\:\mathrm{sin}\left(\mathrm{x}\right)}{\mathrm{1}\:+\:\mathrm{cos}\left(\mathrm{x}\right)}\:\mathrm{dx}\:=\:\mathrm{xtan}\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\:+\:\mathrm{C} \\ $$
Answered by vajpaithegrate@gmail.com last updated on 21/Oct/17
∫((x+2sin (x/2)cos (x/2))/(2cos^2 (x/2)))dx    ∫(x/(2cos^2 (x/2)))+((2sin (x/2)cos (x/2))/(2cos^2 (x/2)))  =∫(x/2)sec^2 (x/2)+tan (x/2)dx  =∫d(xtan ((x/2)))dx  =xtan(x/2)+c  [sin x=2sin (x/2)cos (x/2);1+cos x=2cos^2 (x/2)]
$$\int\frac{\mathrm{x}+\mathrm{2sin}\:\frac{\mathrm{x}}{\mathrm{2}}\mathrm{cos}\:\frac{\mathrm{x}}{\mathrm{2}}}{\mathrm{2cos}\:^{\mathrm{2}} \frac{\mathrm{x}}{\mathrm{2}}}\mathrm{dx}\:\: \\ $$$$\int\frac{\mathrm{x}}{\mathrm{2cos}^{\mathrm{2}} \frac{\mathrm{x}}{\mathrm{2}}}+\frac{\mathrm{2sin}\:\frac{\mathrm{x}}{\mathrm{2}}\mathrm{cos}\:\frac{\mathrm{x}}{\mathrm{2}}}{\mathrm{2cos}\:^{\mathrm{2}} \frac{\mathrm{x}}{\mathrm{2}}} \\ $$$$=\int\frac{\mathrm{x}}{\mathrm{2}}\mathrm{sec}\:^{\mathrm{2}} \frac{\mathrm{x}}{\mathrm{2}}+\mathrm{tan}\:\frac{\mathrm{x}}{\mathrm{2}}\mathrm{dx} \\ $$$$=\int\mathrm{d}\left(\mathrm{xtan}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\right)\mathrm{dx} \\ $$$$=\mathrm{xtan}\frac{\mathrm{x}}{\mathrm{2}}+\mathrm{c} \\ $$$$\left[\mathrm{sin}\:\mathrm{x}=\mathrm{2sin}\:\frac{\mathrm{x}}{\mathrm{2}}\mathrm{cos}\:\frac{\mathrm{x}}{\mathrm{2}};\mathrm{1}+\mathrm{cos}\:\mathrm{x}=\mathrm{2cos}\:^{\mathrm{2}} \frac{\mathrm{x}}{\mathrm{2}}\right] \\ $$
Commented by tawa tawa last updated on 21/Oct/17
God bless you sir
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$

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