What is f(x) = int (x-2)^3 dx if f(2) = -3 ?

1 Answer
Feb 7, 2016

f(x)=1/4x^4-2x^3+6x^2-8x+1

Explanation:

By the Binomial Theorem, (x-2)^3=x^3-6x^2+12x-8

thereforeint(x-2)^3dx=int(x^3-6x^2+12x-8)dx

=1/4x^4-2x^3+6x^2-8x+C.

Now substituting in the initial boundary condition of f(2)=-3, we get

1/4(2)^4-2(2)^3+6(2)^2-8(2)+C=-3

therefore C=-3-4+16-24+16=1.

therefore f(x)=1/4x^4-2x^3+6x^2-8x+1.