Use the power rule for integration:

\(\int x^n\,dx=\dfrac{x^{n+1}}{n+1}+C\).

So

\(\int 6x^2\,dx=6\cdot\dfrac{x^3}{3}+C=2x^3+C\).

Rewrite in powers of \(x\):

\(4\sqrt{x}-\dfrac{3}{x^2}=4x^{1/2}-3x^{-2}\).

Integrate term by term:

\(\int 4x^{1/2}dx=\dfrac{8}{3}x^{3/2}\)

and

\(\int -3x^{-2}dx=3x^{-1}=\dfrac{3}{x}\).

So the answer is

\(\dfrac{8}{3}x^{3/2}+\dfrac{3}{x}+C\).

Integrate term by term:

\(\int x^3dx=\dfrac{x^4}{4}\)

and

\(\int 2x^{-1/2}dx=2\cdot\dfrac{x^{1/2}}{1/2}=4x^{1/2}\).

So the answer is

\(\dfrac{x^4}{4}+4\sqrt{x}+C\).

Integrate each term:

\(\int 12x^2dx=4x^3\)

and

\(\int -5x^{1/2}dx=-5\cdot\dfrac{x^{3/2}}{3/2}=-\dfrac{10}{3}x^{3/2}\).

So

\(Y=4x^3-\dfrac{10}{3}x^{3/2}+C\).

Integrate the gradient:

\(y=\int(3x^2-4)dx=x^3-4x+c\).

Use the point \((2,5)\):

\(5=2^3-4(2)+c=8-8+c\).

So \(c=5\).

Hence

\(y=x^3-4x+5\).