Question

What is `f(x) = int (x-4)^3 dx` if `f(3)=-1`?

Answer 1

`f(x)=((x-4)^4-5)/4`

Explanation:

To integrate, use substitution, then use the rule

`intu^ndu=u^(n+1)/(n+1)+C`

If we set `u=x-4`, then we have `du=dx` and the simplified integral:

`f(x)=intu^3du`

Applying the rule, this becomes

`f(x)=u^4/4+C`

`f(x)=(x-4)^4/4+C`

Now, we can determine `C` since we know that `f(3)=-1`:

`-1=(3-4)^4/4+C`

`=-1=(-1)^4/4+C`

`=-1=1/4+C`

`-5/4=C`

So,

`f(x)=(x-4)^4/4-5/4=((x-4)^4-5)/4`