What is the antiderivative of the function (x-3)*e^x?

We'll integrate by parts. For this reason, we'll consider the
formula:

Int udv = u*v - Int vdu
(*)

We'll put u = x - 3 (1)

We'll
differentiate both sides:

du = dx
(2)

We'll put dv = e^x (3)

We'll
integrate both sides:

Int dv = Int e^x
dx

v = e^x (4)

We'll substitute (1) ,
(2) , (3) and (4) in (*):

Int udv = (x-3)*e^x - Int
(e^x)dx

Int (x-3)*e^x dx = (x-3)*e^x - e^x +
C

Int (x-3)*(e^x)dx = (e^x)*(x-3-1) +
C

The antiderivative of the given function is: Int
(x-3)*(e^x)dx = (e^x)*(x-4) + C