# int x^3e^x dx Find the indefinite integral

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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