intxe^(x-1)dx-intx^2e^-xdx
Let us now deal with only the first integral
intxe^(x-1)dx Let u=x-1 then du=dx
int(u+1)e^udu
Let w=u+1, dw=du , dv=e^u du, v=e^u
(u+1)e^u-inte^udu
(u+1)e^u-e^u+c_1
Substitute back in
xe^(x-1)-e^(x-1)+c_1
Now move onto the second integral
-intx^2e^-xdx
Let u=-x , u^2=x^2 , -du=dx
intu^2e^udu
Let w=u^2 , dw=2udu , dv=e^udu , v=e^u
u^2e^u-2intue^udu
Let w=u , dw=du , dv=e^udu , v=e^u
u^2e^u-2(ue^u-inte^udu)
u^2e^u-2ue^u+2e^u+c_2
Substitute
x^2e^-x+2xe^-x+2e^-x+c_2
Combine both Integrals
xe^(x-1)-e^(x-1)+c_1+ x^2e^-x+2xe^-x+2e^-x+c_2
f(x)=e^(x-1)(x-1)+e^-x(x^2+2x+2)+C
Solve for C
7=e+e^-2(2^2+2(2)+2)+C
7=e+10e^-2+C
C=7-e-10/e^2
f(x)=e^(x-1)(x-1)+e^-x(x^2+2x+2)+7-e-10/e^2