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Bài 1: Nguyên hàm
\(\int {x{e^{2x}}dx} \) 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. \(\dfrac{{{e^{2x}}\left( {2x - 1} \right)}}{4} + C\)
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.
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