Question

Integral x^3 e^-3 ?

Answer 1

`int x^3e^-3 dx= 1/(4e^3)x^4+"c"`

Explanation:

We want to find `intx^3e^-3dx`.

First factor out the constant

`intx^3e^-3dx = e^-3intx^3dx`

Use the power rule to integrate

`intx^ndx=1/(n+1)x^(n+1)+"c"`

`e^-3intx^3dx=e^-3*1/4x^4+"c"=1/(4e^3)x^4+"c"`

Answer 2

`(x^4e^-3)/4+C`

Explanation:

We have:
`intx^3e^-3dx`

Now, notice that `e^-3` is a constant, and therefore can be brought out of the integral.

We now have:

`e^-3intx^3dx`

We use the anti-power rule, which states that:

`intx^ndx=x^(n+1)/(n+1)` where `n` is a constant and `n!=-1`

`intx^-1dx` is equal to `lnabsx`

Using our rule, we have;

`e^-3intx^3dx=e^-3*x^4/4`

`=>(x^4e^-3)/4`

Now, do you `C` why this is incomplete? Right! we need to add `C` to it.

Our answer is:

`(x^4e^-3)/4+C`

Just remember that `e` is a constant- do not treat it as a function!