Tìm nguyên hàm: \(J = \smallint x\ln \frac{{x – 1}}{{x + 1}}dx\)

Đặt \(\left\{ \begin{array}{l}
u = \ln \frac{{x – 1}}{{x + 1}}\\
dv = xdx
\end{array} \right. \Rightarrow \left\{ \begin{array}{l}
du = \frac{2}{{{{\left( {x + 1} \right)}^2}}}dx\\
v = \frac{1}{2}{x^2}
\end{array} \right.\)

Suy ra 

\(\begin{array}{l}
 \Rightarrow I = \frac{1}{2}{x^2}\ln \frac{{x – 1}}{{x + 1}} + \int {\frac{{{x^2}}}{{{{\left( {x + 1} \right)}^2}}}dx = \frac{1}{2}{x^2}\ln \frac{{x – 1}}{{x + 1}} + \int {\left[ {1 – \frac{2}{{x + 1}} + \frac{1}{{{{\left( {x + 1} \right)}^2}}}} \right]dx} } \\
 = \frac{1}{2}{x^2}\ln \frac{{x – 1}}{{x + 1}} + 2 – 2\ln \left| {x – 1} \right| – \frac{1}{{x + 1}} + C
\end{array}\)