# If $f'(x) = 6x^2+2,$ find $f(x)$, given that $f(x) = 7$ when $x=1$

$(A)\; 2x^3+2x+3$
$(B)\; x^3+x+3$
$(C)\; 6x^3+2x+3$
$(D)\; 6x^2+2$

$f'(x) = \frac{d}{dx} (f(x)) = 6x^2 + 2$
Integrating on both sides,
$f(x) = \int (6x^2+2) dx = 2x^3+2x +c$
when $x=1$ and $f(x) = 7$
$\; \; \; \; \; \;7 = 2(1)^3+2(1)+c$
$\implies c = 3$
$\therefore f(x) = 2x^3+2x+3$
answered Dec 20, 2016