∫ e^xcos x dx 

First of all, we have to think of which method we want to use to approach this problem. There are a few options we can consider such as integration by parts and substitution. In this case, integration by parts would be suitable.

Now we have to recall the integration by parts formula which is

∫ u dv/dx dx = uv - ∫ v du/dx dx
From the problem above, 
we can set u= cos x and dv/dx = e^x  

du/dx = -sin x and v= e^x

∫ e^xcosx dx = e^xcos x - ∫ e^x (-sin x) dx
                   = e^xcos x + ∫ e^xsinx dx

Now we have to repeat the integration by parts process again for ∫ e^x sin x dx
Let u= sin x and dv/dx = e^x

       du/dx = cos x and v= e^x

 ∫ e^x cos x dx = e^xcos x + ( e^x sin x - ∫ e^x cos x dx )

2 ∫e^xcos x dx = e^x cos x + e^x sin x
∫  e^x cos x dx = 1/2 ( e^xcos x + e^x sin x )