Calculus of a Single Variable, Chapter 8, 8.2, Section 8.2, Problem 11

Given
int x(e^(-4x)) dx
by applying integration by parts, we'll get the answer
let u=x => u'= 1
v'=e^(-4x) so , v= -1/4e^(-4x)
Now by integration by parts ,
int uv' dx = uv - int u'v dx
so ,
int xe^(-4x) dx = -x/4e^(-4x) -int (1) -1/4e^(-4x) dx
=-x/4e^(-4x) +1/4int e^(-4x) dx
=-x/4e^(-4x) +1/4 int e^(-4x) dx
let us find
int e^(-4x) dx
let u= -4x
du = -4dx so dx = -1/4du
so,
int e^(-4x) dx= int e^(u) -1/4du
=-1/4int e^u du
=-1/4e^u = -1/4e^(-4x)
so, now
int xe^(-4x) dx = -x/4e^(-4x) +1/4int e^(-4x) dx
=-x/4e^(-4x) +1/4 (-1/4)e^(-4x)
=-x/4e^(-4x) -1/16e^(-4x) +C