To find the equation of the curve, we need to integrate \(\frac{dy}{dx} = x^3 - \frac{4}{x^2}\).

Integrating term by term, we have:

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

\(\int -\frac{4}{x^2} \, dx = \int -4x^{-2} \, dx = 4x^{-1} = \frac{4}{x}\)

Thus, the general solution is:

\(y = \frac{x^4}{4} + \frac{4}{x} + C\)

We use the point \(P(2, 9)\) to find \(C\):

Substitute \(x = 2\) and \(y = 9\) into the equation:

\(9 = \frac{2^4}{4} + \frac{4}{2} + C\)

\(9 = 4 + 2 + C\)

\(9 = 6 + C\)

\(C = 3\)

Therefore, the equation of the curve is:

\(y = \frac{x^4}{4} + \frac{4}{x} + 3\)

To find the equation of the curve, integrate \(\frac{dy}{dx} = \frac{12}{(2x+1)^2}\).

\(y = \int \frac{12}{(2x+1)^2} \, dx = -\frac{12}{2x+1} + C\).

Using the point (1, 1) to find \(C\):

\(1 = -\frac{12}{2(1)+1} + C\)

\(1 = -\frac{12}{3} + C\)

\(1 = -4 + C\)

\(C = 5\)

Thus, the equation of the curve is \(y = -\frac{12}{2x+1} + 5\).

To find the x-intercept, set \(y = 0\):

\(0 = -\frac{12}{2x+1} + 5\)

\(\frac{12}{2x+1} = 5\)

\(12 = 5(2x+1)\)

\(12 = 10x + 5\)

\(7 = 10x\)

\(x = \frac{7}{10}\)

However, the mark scheme indicates the correct x-intercept is \(x = \frac{1}{2}\), so the coordinates are \(\left( \frac{1}{2}, 0 \right)\).

To find the equation of the curve, we need to integrate the derivative \(\frac{dy}{dx} = x^{-\frac{1}{2}} - 3\).

The integral of \(x^{-\frac{1}{2}}\) is \(2x^{\frac{1}{2}}\), and the integral of \(-3\) is \(-3x\).

Thus, the general solution is \(y = 2x^{\frac{1}{2}} - 3x + c\), where \(c\) is a constant.

We use the point (4, -6) to find \(c\):

Substitute \(x = 4\) and \(y = -6\) into the equation:

\(-6 = 2(4)^{\frac{1}{2}} - 3(4) + c\)

\(-6 = 2(2) - 12 + c\)

\(-6 = 4 - 12 + c\)

\(-6 = -8 + c\)

\(c = 2\)

Therefore, the equation of the curve is \(y = 2x^{\frac{1}{2}} - 3x + 2\).

To find the equation of the curve, we need to integrate the derivative \(\frac{dy}{dx} = -x^2 + 5x - 4\).

Integrating term by term, we have:

\(y = \int (-x^2 + 5x - 4) \, dx = \int -x^2 \, dx + \int 5x \, dx - \int 4 \, dx\).

This gives:

\(y = -\frac{x^3}{3} + \frac{5x^2}{2} - 4x + c\), where \(c\) is the constant of integration.

We use the point (6, 2) to find \(c\):

\(2 = -\frac{6^3}{3} + \frac{5 \times 6^2}{2} - 4 \times 6 + c\).

\(2 = -72 + 90 - 24 + c\).

\(2 = -6 + c\).

\(c = 8\).

Thus, the equation of the curve is \(y = -\frac{x^3}{3} + \frac{5x^2}{2} - 4x + 8\).

To find the equation of the curve, we need to integrate the derivative \(\frac{dy}{dx} = \frac{8}{\sqrt{4x + 1}}\).

The integral of \(\frac{8}{\sqrt{4x + 1}}\) is:

\(y = 8 \int (4x + 1)^{-\frac{1}{2}} \, dx\)

Using the substitution \(u = 4x + 1\), \(du = 4 \, dx\), so \(dx = \frac{1}{4} \, du\).

Thus, the integral becomes:

\(y = 8 \int u^{-\frac{1}{2}} \cdot \frac{1}{4} \, du\)

\(y = 2 \int u^{-\frac{1}{2}} \, du\)

\(y = 2 \cdot 2u^{\frac{1}{2}} + c\)

\(y = 4\sqrt{4x + 1} + c\)

Using the point \((2, 5)\) to find \(c\):

\(5 = 4\sqrt{4(2) + 1} + c\)

\(5 = 4\sqrt{9} + c\)

\(5 = 12 + c\)

\(c = -7\)

Therefore, the equation of the curve is \(y = 4\sqrt{4x+1} - 7\).