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` .
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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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