Bài 3.12 trang 165 SBT giải tích 12

\(\int {x{e^{2x}}dx} \) bằng

A. \(\dfrac{{{e^{2x}}\left( {x - 2} \right)}}{2} + C\)

B. \(\dfrac{{{e^{2x}} + 1}}{2} + C\)

C. \(\dfrac{{{e^{2x}}\left( {x - 1} \right)}}{2} + C\)

D. \(\dfrac{{{e^{2x}}\left( {2x - 1} \right)}}{4} + C\)

Lời giải

Đặt \(\left\{ \begin{array}{l}u = x\\dv = {e^{2x}}dx\end{array} \right.\) \( \Rightarrow \left\{ \begin{array}{l}du = dx\\v = \dfrac{{{e^{2x}}}}{2}\end{array} \right.\)

Khi đó \(\int {x{e^{2x}}dx}  = \dfrac{{x{e^{2x}}}}{2} - \dfrac{1}{2}\int {{e^{2x}}dx} \) \( = \dfrac{{x{e^{2x}}}}{2} - \dfrac{1}{4}{e^{2x}} + C\) \( = \dfrac{{\left( {2x - 1} \right){e^{2x}}}}{4} + C\).

Chọn D.