Đặt \(\left\{\begin{array}{l} {u=\ln \left(x+\sqrt{1+x^{2} } \right)} \\ {{\rm d}v=\frac{x}{\sqrt{1+x^{2} } } {\rm d}x} \end{array}\right. \, \, \, \, \, \, \Rightarrow \left\{\begin{array}{l} {{\rm d}u=\frac{1}{\sqrt{1+x^{2} } } {\rm d}x} \\ {v=\sqrt{1+x^{2} } } \end{array}\right. .\)
\(\Rightarrow \int \frac{x.\ln \left(x+\sqrt{x^{2} +1} \right)}{\sqrt{x^{2} +1} } {\rm d}x =\sqrt{1+x^{2} } \ln \left(x+\sqrt{1+x^{2} } \right)-\int {\rm d}x \, =\sqrt{1+x^{2} } \ln \left(x+\sqrt{1+x^{2} } \right)-x+C.\)