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

1 Answer
mason m
Mar 17, 2016

f(x)=((x-4)^4-5)/4f(x)=(x−4)4−54

Explanation:

To integrate, use substitution, then use the rule

intu^ndu=u^(n+1)/(n+1)+C∫undu=un+1n+1+C

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

f(x)=intu^3duf(x)=∫u3du

Applying the rule, this becomes

f(x)=u^4/4+Cf(x)=u44+C

f(x)=(x-4)^4/4+Cf(x)=(x−4)44+C

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

-1=(3-4)^4/4+C−1=(3−4)44+C

=-1=(-1)^4/4+C=−1=(−1)44+C

=-1=1/4+C=−1=14+C

-5/4=C−54=C

So,

f(x)=(x-4)^4/4-5/4=((x-4)^4-5)/4f(x)=(x−4)44−54=(x−4)4−54

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