Tìm nguyên hàm của hàm số sau: ∫x2ex/(x+2)2dx

Question

\(\int \frac{x^{2} {\rm e}^{x} }{(x+2)^{2} } {\rm d}x\)

Answer 1

\(J =\int \frac{x^{2} {\rm e}^{x} }{\left(x+2\right)^{2} } {\rm d}x =\int \frac{\left(x^{2} +4x+4-4x-8+4\right){\rm e}^{x} }{\left(x+2\right)^{2} } {\rm d}x  =\int {\rm e}^{x} {\rm d}x-4\int \frac{{\rm e}^{x} }{x+2} {\rm d}x+4\int \frac{{\rm e}^{x} }{\left(x+2\right)^{2} }    {\rm d}x\)

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

\(\Rightarrow  J = {\rm e}^{x} -4\int \frac{{\rm e}^{x} }{x+2} {\rm d}x-\frac{4{\rm e}^{x} }{x+2} +4\int \frac{{\rm e}^{x} }{x+2} {\rm d}x  +C ={\rm e}^{x} -\frac{4{\rm e}^{x} }{x+2} +C.\)