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Question Number 29384 by NECx last updated on 08/Feb/18
Please can it be proven by another  means that          ∫tan^2 xdx=tanx+x +c
$${Please}\:{can}\:{it}\:{be}\:{proven}\:{by}\:{another} \\ $$$${means}\:{that}\: \\ $$$$ \\ $$$$\:\:\:\:\:\int\mathrm{tan}\:^{\mathrm{2}} {xdx}={tanx}+{x}\:+{c} \\ $$
Commented by abdo imad last updated on 08/Feb/18
let put I =∫ tan^2 xdx  the ch.tanx=t give  I= ∫t^2  (dt/(1+t^2 ))=∫ ((1+t^2 −1)/(1+t^2 ))dt=t  −∫  (dt/(1+t^2 )) ⇒  I=t − arctant +λ = tanx −x +λ .
$${let}\:{put}\:{I}\:=\int\:{tan}^{\mathrm{2}} {xdx}\:\:{the}\:{ch}.{tanx}={t}\:{give} \\ $$$${I}=\:\int{t}^{\mathrm{2}} \:\frac{{dt}}{\mathrm{1}+{t}^{\mathrm{2}} }=\int\:\frac{\mathrm{1}+{t}^{\mathrm{2}} −\mathrm{1}}{\mathrm{1}+{t}^{\mathrm{2}} }{dt}={t}\:\:−\int\:\:\frac{{dt}}{\mathrm{1}+{t}^{\mathrm{2}} }\:\Rightarrow \\ $$$${I}={t}\:−\:{arctant}\:+\lambda\:=\:{tanx}\:−{x}\:+\lambda\:. \\ $$
Commented by abdo imad last updated on 08/Feb/18
there is a error in tbe question!
$${there}\:{is}\:{a}\:{error}\:{in}\:{tbe}\:{question}! \\ $$
Commented by NECx last updated on 08/Feb/18
wow.... Thanks
$${wow}….\:{Thanks} \\ $$

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