Tích phân sau \(I = \int\limits_1^e {2x\left( {1 - \ln x} \right)\,dx} \) bằng :

* Hướng dẫn giải

Ta có: \(I = \int\limits_1^e {2x\left( {1 - \ln x} \right)\,dx} \)\(\, = \int\limits_1^e {2x\,dx}  - 2\int\limits_1^e {x\ln \,dx}\)\(\,  = {x^2}\left| {_1^e} \right. - 2\int\limits_1^e {x\ln \,dx} \)

Đặt \({I_1} = \int\limits_1^e {x\ln x\,dx} \)     

Ta có:

\({I_1} = \int\limits_1^e {x\ln x\,dx}  = \left( {\dfrac{{{x^2}}}{2}\ln x} \right)\left| \begin{array}{l}^e\\_1^{}\end{array} \right. - \int\limits_1^e {\dfrac{x}{2}dx} \)

\(= \left( {\dfrac{{{x^2}}}{2}\ln x} \right)\left| \begin{array}{l}^e\\_1^{}\end{array} \right. - \left( {\dfrac{{{x^2}}}{4}} \right)\left| \begin{array}{l}_{}^e\\_1^{}\end{array} \right.\)

\( = \dfrac{e^2}{2}\ln e - \left( {\dfrac{e^2}{4} - \dfrac{1}{4}} \right) = \dfrac{e^2}{2}+\dfrac {1}{4}\)

Khi đó ta có: \(I = {e^2} - 1 - 2.\left( {\dfrac{{{e^2}}}{4} + \dfrac{1}{4}} \right) = \dfrac{{{e^2} - 3}}{2}\)