Question

What is `f(x) = int (x-3)^2-3x+4 dx` if `f(2) = 1`?

Answer 1

`f(x) = x^3/3 -(9x^2)/2 +13x -29/3`

Explanation:

First, expand the integrant as follow

`int((x-3)^2 -3x +4)dx = int(x^2-6x+9-3x+4)dx`

`=int(x^2-9x+13)dx`

Then we can integrate this using the power rule like this

`f(x) = x^3/3-(9x^2)/2+13x+C`#

We are given `f(2) = 1` , substitute this into the `f(x)` to solve for C

`1= (2^3)/3-(9(2)^2)/2+13(2) +C`

`1= 8/3 -36/2 +26+C`

`1= 8/3 -18 +26+C`

`C= -29/3`

So `f(x) = x^3/3 -(9x^2)/2 +13x -29/3`