Calculus of a Single Variable, Chapter 8, 8.2, Section 8.2, Problem 13

To evaluate the integral: int x^3e^x dx , we may apply "integration by parts": int u *dv = uv- int vdu .
Let: u= x^3 then du = 3x^2 dx
dv = e^x dx then v = int e^x dx = e^x .

Apply the formula for integration by parts, we get:
int x^3e^x dx = x^3 e^x - int 3x^2e^xdx .
= x^3 e^x - 3 int x^2e^xdx.
To evaluate int x^2 e^x dx , we apply another set of integration by parts.
Let: u = x^2 then du = 2x dx
v=e^x dx then dv = e^x
The integral becomes:
int x^2 e^x dx =x^2e^x - int 2xe^x dx
Another set of integration by parts by letting:
u = 2x then du =2dx
v=e^x dx then dv = e^x
int 2xe^x dx = 2xe^x - int 2e^x dx
= 2xe^x -2 e^x +C
Using int 2xe^x dx =2xe^x - 2e^x +C , we get:
int x^2 e^x dx =x^2e^x - int 2xe^x dx
=x^2e^x - [2xe^x - 2e^x ]+C
=x^2e^x - 2xe^x + 2e^x +C
Then,
int x^3e^x dx = x^3 e^x - 3 int x^2e^xdx .
= x^3 e^x - 3 [x^2e^x - 2xe^x + 2e^x] +C
= x^3 e^x - 3x^2e^x +6xe^x -6 e^x +C