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)#
