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`

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