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Question Number 169250 by mathocean1 last updated on 26/Apr/22

Calculate B=∫(1/(1+sin(x)))dx using u=tan((x/2))

$${Calculate}\:{B}=\int\frac{\mathrm{1}}{\mathrm{1}+{sin}\left({x}\right)}{dx}\:{using}\:{u}={tan}\left(\frac{{x}}{\mathrm{2}}\right) \\ $$

Commented by infinityaction last updated on 27/Apr/22

   I = ∫(dx/((sin (x/2) + cos (x/2))^2 ))    I = ∫((sec^2 x/2)/((1+tan x/2)^2 ))dx     u = tan x/2     du = (1/2)sec^2 x/2     I  = ∫((2du)/((1+u)^2 ))     I   =  −(2/(1+u)) + c      I   = −(2/(1+tan x/2)) + c

$$\:\:\:{I}\:=\:\int\frac{{dx}}{\left(\mathrm{sin}\:\frac{{x}}{\mathrm{2}}\:+\:\mathrm{cos}\:\frac{{x}}{\mathrm{2}}\right)^{\mathrm{2}} } \\ $$$$\:\:{I}\:=\:\int\frac{\mathrm{sec}\:^{\mathrm{2}} {x}/\mathrm{2}}{\left(\mathrm{1}+\mathrm{tan}\:{x}/\mathrm{2}\right)^{\mathrm{2}} }{dx} \\ $$$$\:\:\:{u}\:=\:\mathrm{tan}\:{x}/\mathrm{2} \\ $$$$\:\:\:{du}\:=\:\frac{\mathrm{1}}{\mathrm{2}}\mathrm{sec}\:^{\mathrm{2}} {x}/\mathrm{2} \\ $$$$\:\:\:{I}\:\:=\:\int\frac{\mathrm{2}{du}}{\left(\mathrm{1}+{u}\right)^{\mathrm{2}} } \\ $$$$\:\:\:{I}\:\:\:=\:\:−\frac{\mathrm{2}}{\mathrm{1}+{u}}\:+\:{c} \\ $$$$\:\:\:\:{I}\:\:\:=\:−\frac{\mathrm{2}}{\mathrm{1}+\mathrm{tan}\:{x}/\mathrm{2}}\:+\:{c} \\ $$

Answered by MikeH last updated on 27/Apr/22

sin x = ((2t)/(1+t^2 ))  dx = ((2dt)/(1+t^2 ))  ⇒ B = ∫(1/((1+((2t)/(1+t^2 ))))).((2dt)/(1+t^2 )) = ∫((2dt)/(t^2 +2t+1))  B = 2∫(dt/((1+t)^2 )) = −(2/(1+t)) + k  B = −(2/(1+tan((x/2))))+k

$$\mathrm{sin}\:{x}\:=\:\frac{\mathrm{2}{t}}{\mathrm{1}+{t}^{\mathrm{2}} } \\ $$$${dx}\:=\:\frac{\mathrm{2}{dt}}{\mathrm{1}+{t}^{\mathrm{2}} } \\ $$$$\Rightarrow\:{B}\:=\:\int\frac{\mathrm{1}}{\left(\mathrm{1}+\frac{\mathrm{2}{t}}{\mathrm{1}+{t}^{\mathrm{2}} }\right)}.\frac{\mathrm{2}{dt}}{\mathrm{1}+{t}^{\mathrm{2}} }\:=\:\int\frac{\mathrm{2}{dt}}{{t}^{\mathrm{2}} +\mathrm{2}{t}+\mathrm{1}} \\ $$$${B}\:=\:\mathrm{2}\int\frac{{dt}}{\left(\mathrm{1}+{t}\right)^{\mathrm{2}} }\:=\:−\frac{\mathrm{2}}{\mathrm{1}+{t}}\:+\:{k} \\ $$$${B}\:=\:−\frac{\mathrm{2}}{\mathrm{1}+\mathrm{tan}\left(\frac{{x}}{\mathrm{2}}\right)}+{k} \\ $$

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