Question

Calculate f(x)?

`int_0^1x^2e^(-x)dx`

Answer 1

`~~0.16`

Explanation:

1. Find the integral of `x^2*e^(-x)`
   Integration by parts:

`int(f*dg)dx=f*g-int(g*df)dx`
Let `x^2=f` and `e^(-x)=dg`
Therefore, `g=-e^-x` and `df=2x`
`int(x^2*e^(-x))dx=-x^2e^-x-int(-2x*e^-x)dx`

Again, we've got to use integration by parts for the integral of `-2x*e^-x`

`int(-2x*e^-x)dx`
`f=-2x`
`df=-2`
`g=-e^-x`
`dg=e^-x`
`int(-2x*e^-x)dx=2x*e^-x-int(2e^-x)dx`
`int(2e^-x)dx=-2e^-x`

All in all we get the equation

`int(x^2*e^(-x))dx=-x^2e^-x-(2x*e^-x-(-2e^-x))`
`int(x^2*e^(-x))dx=-x^2e^-x-2x*e^-x-2e^-x`

1. Now we calculate the value by inserting the boundaries.
   `[-x^2e^-x-2x*e^-x-2e^-x]_0^1`
   `(-1e^-1-2e^-1-2e^-1)-(0-0-2e^0)`
   `-5e^-1+2~~0.16`