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

##### 1 Answer
Jan 24, 2016

$f \left(x\right) = {x}^{3} / 3 - \frac{9 {x}^{2}}{2} + 13 x - \frac{29}{3}$

#### Explanation:

First, expand the integrant as follow

$\int \left({\left(x - 3\right)}^{2} - 3 x + 4\right) \mathrm{dx} = \int \left({x}^{2} - 6 x + 9 - 3 x + 4\right) \mathrm{dx}$

$= \int \left({x}^{2} - 9 x + 13\right) \mathrm{dx}$

Then we can integrate this using the power rule like this

$f \left(x\right) = {x}^{3} / 3 - \frac{9 {x}^{2}}{2} + 13 x + C$#

We are given $f \left(2\right) = 1$ , substitute this into the $f \left(x\right)$ to solve for C

$1 = \frac{{2}^{3}}{3} - \frac{9 {\left(2\right)}^{2}}{2} + 13 \left(2\right) + C$

$1 = \frac{8}{3} - \frac{36}{2} + 26 + C$

$1 = \frac{8}{3} - 18 + 26 + C$

$C = - \frac{29}{3}$

So $f \left(x\right) = {x}^{3} / 3 - \frac{9 {x}^{2}}{2} + 13 x - \frac{29}{3}$