Tính tích phân \(I = \int\limits_0^1 {3x.{e^{2x}}} dx.\)

Tính tích phân \(I = \int\limits_0^1 {3x.{e^{2x}}} dx.\)
A. \(I = \frac{{3{e^2} + 3}}{{16}}\)
B. \(I = \frac{{2{e^2} + 2}}{9}\)
C. \(I = \frac{{3{e^2} + 3}}{4}\)
D. \(I = \frac{{2{e^2} + 2}}{3}\)

Đặt \(\left\{ {\begin{array}{*{20}{c}}{u = 3x}\\{dv = {e^{2x}}dx}\end{array}} \right. \Rightarrow \left\{ {\begin{array}{*{20}{c}}{du = 3dx}\\{v = \frac{{{e^{2x}}}}{2}}\end{array}} \right. \Rightarrow I = \frac{{3x.{e^{2x}}}}{2}\left| {\begin{array}{*{20}{c}}1\\0\end{array}} \right. - \frac{3}{2}\int\limits_0^1 {{e^{2x}}dx} = \frac{{3x.{e^{2x}}}}{2}\left| {\begin{array}{*{20}{c}}1\\0\end{array}} \right. - \frac{3}{4}{e^{2x}}\left| {\begin{array}{*{20}{c}}1\\0\end{array} = \frac{{3{e^2} + 3}}{4}} \right..\)