1. ## integrate [E^(2x)](Cos[2x])

2. Originally Posted by guess
$
e^{2x} \cos(2x)= \mbox{Re}\ (e^{2x} e^{2x \mathrm{i}})=\mbox{Re}\ (e^{2x(1+\mathrm{i})})
$

Hence:

$
\int e^{2x} \cos(2x) dx= \int \mbox{Re}\ (e^{2x(1+\mathrm{i})})dx=\mbox{Re}\ \left(\int e^{2x(1+\mathrm{i})}dx \right)
$

$
\int e^{2x} \cos(2x) dx= \mbox{Re}\ \left(\frac{e^{2x(1+\mathrm{i})}}{2(1+\mathrm{i})} \right) + C
$
.

You should be able to complete the simplification from here.

RonL

3. Originally Posted by guess
You have,
$\int e^{2x}\cos 2xdx$
Integrate by parts, $u'=e^{2x}\mbox{ and } v=\cos 2x$
$\frac{1}{2}e^{2x}\cos 2x+\int e^{2x}\sin 2x dx$
Integrate by parts, $u'=e^{2x}\mbox{ and } v=\sin 2x$
$\frac{1}{2}e^{2x}\cos 2x+\left( \frac{1}{2}e^{2x}\sin 2x-\int e^{2x}\cos 2x dx \right)$.
Thus,
$\int e^{2x}\cos 2xdx=\frac{1}{2}e^{2x}\cos 2x+\left( \frac{1}{2}e^{2x}\sin 2x-\int e^{2x}\cos 2x dx \right)$
Thus, (combine integrals),
$2\int e^{2x}\cos 2xdx=\frac{1}{2}e^{2x}\cos 2x+ \frac{1}{2}e^{2x}\sin 2x$
Thus,
$\int e^{2x}\cos 2xdx=\frac{1}{4}e^{2x}(\sin 2x+\cos 2x)+C$

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### integration of e^2x cos2x

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