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

Expert Answers

An illustration of the letter 'A' in a speech bubbles

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`

See eNotes Ad-Free

Start your 48-hour free trial to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Get 48 Hours Free Access
Approved by eNotes Editorial