$\displaystyle \int e^x(\cos{x})dx$
substituted it to
dv = e^(x) dx
v = e^(x)
u = (cos(x))
u = -sin(x) dx
and this one goes infinity for me
help needed for what to sub ... i dunno how the integrator got a short answer
Give this integral a name,
Let $\displaystyle X=\int e^x \cos x dx$
After doing integration by parts you get,
$\displaystyle X = \mbox{ something } + \int e^x \sin x dx$
After doing integration by parts again you get,
$\displaystyle X = \mbox{ something } + \mbox{ more something }- \int e^x \cos x dx$
And it seems it goes on forever. But $\displaystyle \int e^x \cos x dx = X$
So,
$\displaystyle X = \mbox{ something } + \mbox{ more something } - X$
Thus,
$\displaystyle 2X = \mbox{ something } + \mbox{ more something }$
Thus,
$\displaystyle X = \frac{ \mbox{ something }+\mbox{ more something }}{2} + C$