What is the antiderivative of the function f(x) given by f(x)=x*e^8x ?

We'll integrate by parts, according to the identity
below:

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

We'll put u = x. (1)

We'll
differentiate both sides:

du = dx
(2)

We'll put dv = e^8x (3)

We'll
integrate both sides:

Int dv = Int e^8x
dx

v = e^8x/8 (4)

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

Int udv = x*e^8x/8 - Int
(e^8x/8)dx

Int (x*e^8x)dx = x*e^8x/8 - e^8x/64 +
C