Question Number 46 by surabhi last updated on 25/Jan/15
$$\int\left(\mathrm{log}\:{x}\right)^{\mathrm{2}} {dx} \\ $$
Answered by surabhi last updated on 04/Nov/14
![∫(log x)^2 dx=∫∣(log x)^2 ∙1∣dx =(log x)^2 ∙∫1 dx−∫{(d/dx)(log x)^2 ∙∫1 dx}dx =x(log x)^2 −∫(((2log x)/x)∙x)dx =x(log x)^2 −2∫(log x∙1)dx =x(log x)^2 −2[(log x)∫dx−∫{(d/dx)(log x)∙∫dx}dx] =x(log x)^2 −2[x log x−∫(1/x)∙x dx] =x(log x)^2 −2x log x+2x+C](https://www.tinkutara.com/question/Q47.png)
$$\int\left(\mathrm{log}\:{x}\right)^{\mathrm{2}} {dx}=\int\mid\left(\mathrm{log}\:{x}\right)^{\mathrm{2}} \centerdot\mathrm{1}\mid{dx} \\ $$$$=\left(\mathrm{log}\:{x}\right)^{\mathrm{2}} \centerdot\int\mathrm{1}\:{dx}−\int\left\{\frac{{d}}{{dx}}\left(\mathrm{log}\:{x}\right)^{\mathrm{2}} \centerdot\int\mathrm{1}\:{dx}\right\}{dx} \\ $$$$={x}\left(\mathrm{log}\:{x}\right)^{\mathrm{2}} −\int\left(\frac{\mathrm{2log}\:{x}}{{x}}\centerdot{x}\right){dx} \\ $$$$={x}\left(\mathrm{log}\:{x}\right)^{\mathrm{2}} −\mathrm{2}\int\left(\mathrm{log}\:{x}\centerdot\mathrm{1}\right){dx} \\ $$$$={x}\left(\mathrm{log}\:{x}\right)^{\mathrm{2}} −\mathrm{2}\left[\left(\mathrm{log}\:{x}\right)\int{dx}−\int\left\{\frac{{d}}{{dx}}\left(\mathrm{log}\:{x}\right)\centerdot\int{dx}\right\}{dx}\right] \\ $$$$={x}\left(\mathrm{log}\:{x}\right)^{\mathrm{2}} −\mathrm{2}\left[{x}\:\mathrm{log}\:\:{x}−\int\frac{\mathrm{1}}{{x}}\centerdot{x}\:{dx}\right] \\ $$$$={x}\left(\mathrm{log}\:{x}\right)^{\mathrm{2}} −\mathrm{2}{x}\:\mathrm{log}\:{x}+\mathrm{2}{x}+{C} \\ $$