A curve has a gradient function of $\displaystyle 3x^2-2x+5$.Given that the curve passes through the point $\displaystyle (0,2)$,find the equation of the curve.
please show me the step by step..
This isn't finding an area, it's finding an antiderivative...
The function you are looking for is:
$\displaystyle f(x) = \int{3x^2 - 2x + 5\,dx}$
$\displaystyle = x^3 - x^2 + 5x + C$.
Now using the fact that $\displaystyle (0, 2)$ lies on the curve:
$\displaystyle 2 = 0^3 - 0^2 + 5\cdot 0 + C$
$\displaystyle 2 = C$.
So the function is
$\displaystyle f(x) = x^3 - x^2 + 5x + 2$.