\(I=\int cos2x.e^{3x}dx\)
Đặt \(\left\{{}\begin{matrix}u=e^{3x}\\dv=cos2xdx\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}du=3.e^{3x}dx\\v=\dfrac{1}{2}sin2x\end{matrix}\right.\)
\(\Rightarrow I=\dfrac{1}{2}e^{3x}.sin2x-\dfrac{3}{2}\int sin2x.e^{3x}dx=\dfrac{1}{2}e^{3x}.sin2x-\dfrac{3}{2}I_1\)
Xét \(I_1=\int sin2x.e^{3x}dx\) \(\Rightarrow\) đặt \(\left\{{}\begin{matrix}u=e^{3x}\\dv=sin2x.dx\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}du=3.e^{3x}dx\\v=\dfrac{-1}{2}cos2x\end{matrix}\right.\)
\(\Rightarrow I_1=-\dfrac{1}{2}cos2x.e^{3x}+\dfrac{3}{2}\int cos2x.e^{3x}dx\) \(=-\dfrac{1}{2}cos2x.e^{3x}+\dfrac{3}{2}I\)
\(\Rightarrow I=\dfrac{1}{2}e^{3x}sin2x-\dfrac{3}{2}\left(-\dfrac{1}{2}e^{3x}cos2x+\dfrac{3}{2}I\right)\)
\(\Rightarrow I=\dfrac{1}{2}e^{3x}sin2x+\dfrac{3}{4}e^{3x}cos2x-\dfrac{9}{4}I\)
\(\Rightarrow\dfrac{13}{4}I=\dfrac{1}{2}e^{3x}sin2x+\dfrac{3}{4}e^{3x}cos2x\)
\(\Rightarrow I=\dfrac{2}{13}e^{3x}sin2x+\dfrac{3}{13}e^{3x}cos2x+C\)
