Tìm nguyên hàm của hàm số \(f\left( x \right) = x\ln \left( {x + 2} \right)\).

Tìm nguyên hàm của hàm số \(f\left( x \right) = x\ln \left( {x + 2} \right)\).
A. \(\int {f\left( x \right){\rm{d}}x} = \frac{{{x^2}}}{2}\ln \left( {x + 2} \right) - \frac{{{x^2} + 4x}}{4} + C\).
B. \(\int {f\left( x \right){\rm{d}}x} = \frac{{{x^2} - 4}}{2}\ln \left( {x + 2} \right) - \frac{{{x^2} - 4x}}{4} + C\).
C. \(\int {f\left( x \right){\rm{d}}x} = \frac{{{x^2}}}{2}\ln \left( {x + 2} \right) - \frac{{{x^2} + 4x}}{2} + C\).
D. \(\int {f\left( x \right){\rm{d}}x} = \frac{{{x^2} - 4}}{2}\ln \left( {x + 2} \right) - \frac{{{x^2} + 4x}}{2} + C\).

Đặt \(\left\{ \begin{array}{l}u = \ln \left( {x + 2} \right)\\{\rm{d}}v = x{\rm{d}}x\end{array} \right. \Rightarrow \left\{ \begin{array}{l}{\rm{d}}u = \frac{{{\rm{d}}x}}{{x + 2}}\\v = \frac{{{x^2}}}{2}\end{array} \right.\).
Ta có:
\(\begin{array}{l}\int {f\left( x \right){\rm{d}}x} = \frac{{{x^2}}}{2}\ln \left( {x + 2} \right) - \frac{1}{2}\int {\frac{{{x^2}}}{{x + 2}}{\rm{d}}x} = \frac{{{x^2}}}{2}\ln \left( {x + 2} \right) - \frac{1}{2}\int {\left( {x - 2 + \frac{4}{{x + 2}}} \right){\rm{d}}x} \\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{{x^2}}}{2}\ln \left( {x + 2} \right) - \frac{1}{2}\left( {\frac{{{x^2}}}{2} - 2x + 4\ln \left( {x + 2} \right)} \right) + C\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{{x^2} - 4}}{2}\ln \left( {x + 2} \right) - \frac{{{x^2} - 4x}}{4} + C.\end{array}\)