Question

evaluate the integrals by integrating by parts:

Answer 1

\[\int\limits_{}e^x \cos(2x) {dx}\]

Answer 2

\[I=\int\limits_{ }^{}e^{x} \cos2x dx= \cos2x \int\limits_{}^{} e^{x}dx - \int\limits_{}^{} [d(\cos2x)/dx . \int\limits_{}^{} e^{x}dx]dx\]
\[= \cos2x e^{x} + 2\int\limits_{}^{} \sin2xe^{x}dx\]\[=\cos2xe^{x} + 2\left( \sin2x \int\limits_{}^{}e^{x}dx - \int\limits_{}^{} [d(\sin2x)/dx .\int\limits_{}^{} e^{x}dx]dx{} \right)\]\[=\cos2x e^{x} + 2 \sin2x e^{x} -4\int\limits_{}^{}\cos2xe^{x}dx\]
\[I = e^{x}(\cos2x+2\sin2x)- 4I\]\[I =e^{x}(\cos2x + 2\sin2x) /5\]