Tìm nguyên hàm \(J = \smallint \frac{{\left( {\ln x + 1} \right)\ln x}}{{{{\left( {\ln x + 1 + x} \right)}^3}}}dx\)

Ta có : \(J = \smallint \frac{{\ln x + 1}}{{x{{\left( {\frac{{\ln x + 1}}{x} + 1} \right)}^3}}}.\frac{{\ln x}}{{{x^2}}}dx\)

Đặt \(t = \frac{{\ln x + 1}}{x} \Rightarrow dt =  – \frac{{\ln x}}{{{x^2}}}dx\)

Suy ra 

\(\begin{array}{l}
J =  – \int {\frac{{tdt}}{{{{\left( {t + 1} \right)}^3}}}}  = \int {\left[ {\frac{1}{{{{\left( {t + 1} \right)}^3}}} – \frac{1}{{{{\left( {t + 1} \right)}^2}}}} \right]dt} \\
 =  – \frac{1}{{2{{\left( {t + 1} \right)}^2}}} + \frac{1}{{t + 1}} + C\\
 =  – \frac{{{x^2}}}{{2\left( {\ln x + 1 + {x^2}} \right)}} + \frac{x}{{\ln x + x + 1}} + C
\end{array}\)