1. int by parts(almost finished)

original problem:
$\displaystyle \int e^xcos(2x)dx$

I've used int. by parts twice so far and I have this:
$\displaystyle e^xcos(2x) + 2e^xsin(2x) - 4\int e^xcos(2x)dx$

Where do I go from here? I'm lost...

2. Originally Posted by saiyanmx89
original problem:
$\displaystyle \int e^xcos(2x)dx$

I've used int. by parts twice so far and I have this:
$\displaystyle e^xcos(2x) + 2e^xsin(2x) - 4\int e^xcos(2x)dx$

Where do I go from here? I'm lost...
$\displaystyle \int e^xcos(2x)dx= e^xcos(2x) + 2e^xsin(2x) - 4\int e^xcos(2x)dx$

so

$\displaystyle \int 5e^xcos(2x)dx= e^xcos(2x) + 2e^xsin(2x)$

so

$\displaystyle \int e^xcos(2x)dx= \frac{e^xcos(2x) + 2e^xsin(2x)}{5}$

3. where did you get 5e^x from? that's where I get lost....

4. $\displaystyle \textcolor{red}{\int e^xcos(2x)dx}= e^xcos(2x) + 2e^xsin(2x) \textcolor{red}{- 4\int e^xcos(2x)dx}$

note the like terms ... now move $\displaystyle \textcolor{red}{- 4\int e^xcos(2x)dx}$ to the left side and combine.