How do you integrate #int e^xcosx# by integration by parts method?
By integrating by parts twice:
We will use the fact that
In particular, we're interested in the fact that deriving the cosine function twice means to invert its sign.
So, after a first integration by parts, we have
Putting all the pieces together, we have
As you can see, the (same) integral appears on both sides, this means that we can add it to both sides to get
and thus solve for it:
Integrate by parts again:
The integral now appears on both sides of the equation and we can solve for it: