Application for integrals

Exercise:

If $\int{f'\left( x \right)dx}={{x}^{2}}+3x+3$ and $f\left( 1 \right)=2$

Determine $f\left( 8 \right)$

Solution: we have $\int{f'\left( x \right)dx}={{x}^{2}}+3x+3$

$\Rightarrow f\left( x \right)={{x}^{2}}+3x+3+c$  

but $f\left( 1 \right)=2\Rightarrow 2=1+3+3+c\Rightarrow 2-7=c\Rightarrow c=-5$

Thus $f\left( x \right)={{x}^{2}}+3x-2$ so $f\left( 8 \right)={{8}^{2}}+3\left( 8 \right)-2=64+24-2=86$