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

1 Answer
Mar 17, 2016

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