\(I=\int e^xcos2xdx\)
Đặt \(\left\{{}\begin{matrix}u=e^x\\dv=cos2xdx\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}du=e^xdx\\v=\dfrac{1}{2}sin2x\end{matrix}\right.\)
\(\Rightarrow I=\dfrac{1}{2}e^xsin2x-\dfrac{1}{2}\int e^xsin2xdx\)
Xét \(I_1=\int e^xsin2xdx\)
Đặt \(\left\{{}\begin{matrix}u=e^x\\dv=sin2xdx\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}du=e^xdx\\v=-\dfrac{1}{2}cos2x\end{matrix}\right.\)
\(\Rightarrow I_1=-\dfrac{1}{2}e^xcos2x+\dfrac{1}{2}\int e^xcos2xdx=-\dfrac{1}{2}e^xcos2x+\dfrac{1}{2}I\)
\(\Rightarrow I=\dfrac{1}{2}e^xsin2x-\dfrac{1}{2}\left(-\dfrac{1}{2}e^xcos2x+\dfrac{1}{2}I\right)\)
\(\Rightarrow\dfrac{5}{4}I=\dfrac{1}{2}e^xsin2x+\dfrac{1}{4}e^xcos2x+C\)
\(\Rightarrow I=\dfrac{2}{5}e^xsin2x+\dfrac{1}{5}e^xcos2x+C\)