Question and Answers Forum

All Questions      Topic List

Integration Questions

Previous in All Question      Next in All Question      

Previous in Integration      Next in Integration      

Question Number 155897 by cortano last updated on 05/Oct/21

∫ x(arctan x)^2 dx=?

$$\:\int\:\mathrm{x}\left(\mathrm{arctan}\:\mathrm{x}\right)^{\mathrm{2}} \:\mathrm{dx}=? \\ $$

Answered by puissant last updated on 05/Oct/21

Q=∫x(arctanx)^2 dx.. IBP ⇒ .. { ((u=arctan x)),((v′=xarctan x)) :} ⇒ { ((u′=(1/(1+x^2 )))),((v=−(x/2)+(1/2)(1+x^2 )arctanx)) :} ⇒ Q=−(x/2)arctanx+(1/2)(1+x^2 )arctan^2 x−(1/2)∫{arctanx+(x/(1+x^2 ))}dx =−(x/2)arctanx+(1/2)(1+x^2 )arctan^2 x−(1/2)(xarctanx−(1/2)ln(1+x^2 ))+(1/4)ln(1+x^2 )+C ∴∵ Q=(1/2)(1+x^2 )arctanx−x arctanx+(1/2)ln(1+x^2 )+C... ...........................Le puissant........................

$${Q}=\int{x}\left({arctanx}\right)^{\mathrm{2}} {dx}.. \\ $$$${IBP}\:\:\Rightarrow\:.. \\ $$$$\begin{cases}{{u}={arctan}\:{x}}\\{{v}'={xarctan}\:{x}}\end{cases}\:\:\Rightarrow\:\:\begin{cases}{{u}'=\frac{\mathrm{1}}{\mathrm{1}+{x}^{\mathrm{2}} }}\\{{v}=−\frac{{x}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}+{x}^{\mathrm{2}} \right){arctanx}}\end{cases} \\ $$$$\Rightarrow\:{Q}=−\frac{{x}}{\mathrm{2}}{arctanx}+\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}+{x}^{\mathrm{2}} \right){arctan}^{\mathrm{2}} {x}−\frac{\mathrm{1}}{\mathrm{2}}\int\left\{{arctanx}+\frac{{x}}{\mathrm{1}+{x}^{\mathrm{2}} }\right\}{dx} \\ $$$$=−\frac{{x}}{\mathrm{2}}{arctanx}+\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}+{x}^{\mathrm{2}} \right){arctan}^{\mathrm{2}} {x}−\frac{\mathrm{1}}{\mathrm{2}}\left({xarctanx}−\frac{\mathrm{1}}{\mathrm{2}}{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\right)+\frac{\mathrm{1}}{\mathrm{4}}{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)+{C} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\therefore\because\:\:\:{Q}=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}+{x}^{\mathrm{2}} \right){arctanx}−{x}\:{arctanx}+\frac{\mathrm{1}}{\mathrm{2}}{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)+{C}... \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...........................\mathscr{L}{e}\:{puissant}........................ \\ $$

Commented by SANOGO last updated on 06/Oct/21

le dur gar

$${le}\:{dur}\:{gar} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com