We need to find the antiderivative of f(x)=x*e^4x

We solve this problem using Integration by parts:

Int [f(x)g'(x) dx = f(x)g(x)- Int [ f'(x)g(x) dx]

Let f(x) = x and g'(x)= e^4x

=> g(x) = e^4x/4

Int [ x*e^4x dx] = x*e^4x / 4 - Int [ e^4x/4 dx]

=> x* e^4x/4 - e^4x / 16 + C

**Therefore the antiderivative of f(x)=x*e^4x is x*e^4x/4 - e^4x /16 + C**

To determine what is requested by enunciation, we'll have to evaluate the indefinite integral of the given function.

Int f(x)dx

We'll integrate by parts, so, we'll recall the formula:

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^4x (3)

We'll integrate both sides:

Int dv = Int e^4x dx

v = e^4x/4 (4)

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

Int udv = x*e^4x/4 - Int (e^4x/4)dx

**The anti-derivative is: **

**Int (x*e^4x)dx = (x*e^4x)/4 - (e^4x)/16 + C**