f(x) = int (x e^x - x)dx
=>f(x) = int xe^xdx - int x dx
=>f(x) = int x e^x dx - 1/2 x^2 + C_2
where C_2 is an arbitrary constant of integration.
To solve the first integral, we will employ integration by parts. Let u equiv x and dv equiv e^xdx.
int x e^x dx = uv - int v du
=>int x e^x dx = xe^x - int e^x dx + C_1
=>int xe^x dx = xe^x - e^x + C_1
=>int xe^x dx = color(blue)(e^x(x - 1) + C_1)
where C_1 is an arbitrary constant of integration.
We can now substitute this result back into our f(x).
f(x) = color(blue)(e^x(x - 1) + C_1) - 1/2x^2 + C_2
=>f(x) = e^x(x-1)-1/2x^2 + C
where C is an arbitrary constant of integration (C_1 and C_2 were arbitrary, so they were combined into a single constant term).
Assessing at f(0):
f(0) = -2 = e^0(0-1)-1/2(0)^2+C
=>-2 = -1 +C
=>C = -1
Hence, the final result is:
=>color(green)(f(x) = e^x(x-1)-1/2x^2 -1)