OCR MEI C2 (Core Mathematics 2)

Find \(\int 7x^2 dx\). [3]

The gradient of a curve is given by \(\frac{dy}{dx} = \frac{18}{x} + 2\). The curve passes through the point \((3, 6)\). Find the equation of the curve. [5]

The gradient of a curve is given by \(\frac{dy}{dx} = 6x^{\frac{1}{2}} - 5\). Given also that the curve passes through the point \((4, 20)\), find the equation of the curve. [5]

Find \(\int_2^5 (2x^3 + 3) dx\). [3]

The gradient of a curve is given by \(\frac{dy}{dx} = 6\sqrt{x} - 2\). Given also that the curve passes through the point \((9, 4)\), find the equation of the curve. [5]

Find \(\int_2^5 \left(1 - \frac{6}{x^3}\right) dx\). [4]

Find \(\int_1^2 (12x^5 + 5) dx\). [4]

The gradient of a curve is \(3\sqrt{x} - 5\). The curve passes through the point \((4, 6)\). Find the equation of the curve. [5]

A curve has gradient given by \(\frac{dy}{dx} = 6\sqrt{x}\). Find the equation of the curve, given that it passes through the point \((9, 105)\). [4]

Find \(\int_1^2 \left(x^4 - \frac{3}{x^2} + 1\right) dx\), showing your working. [5]

Find \(\int 30x^2 dx\). [3]

Find \(\int (x^5 + 10x^3) dx\). [4]