You would need to do a bit of integration by parts here.
lets first set f(x) = x^2 and g'(x) = e^(x-1)
and use u = f(x) and v = g(x)
then du = f'(x) dx and dv = g'(x)
and we use the integral by parts formula:
intudv = uv - intv du
Now let us get the values:
u = x^2 and v= e^(x-1)
du = 2x and dv= e^(x-1)
Thus:
int x^2e^(x-1) dx = x^2e^(x-1) - int 2xe^(x-1) dx
int x^2e^(x-1) dx = x^2e^(x-1) - 2int xe^(x-1) dx
now lets solve for color(red)(int xe^(x-1) dx)
color(red)(u = x) and color(red)(v=e^(x-1))
color(red)(du = 1) and color(red)(dv=e^(x-1))
Thus:
color(red)( int xe^(x-1) = xe^(x-1) - int 1e^(x-1)
which equals:
color(red)( int xe^(x-1) = xe^(x-1) - e^(x-1)
Now we can substitute that back into our first problem, and get:
int x^2e^(x-1) dx = x^2e^(x-1) - 2int xe^(x-1) dx
is equal to, int x^2e^(x-1) dx = x^2e^(x-1) - 2(xe^(x-1) - e^(x-1))
where we can simplify to:
int x^2e^(x-1) = e^(x-1)(x^2 - 2x + 2)
