OCR MEI C2 (Core Mathematics 2)

1 Find \(\int 7 x ^ { \frac { 5 } { 2 } } \mathrm {~d} x\).

2 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 18 } { x ^ { 3 } } + 2\). The curve passes through the point \(( 3,6 )\). Find the
equation of the curve. equation of the curve.

3 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { \frac { 1 } { 2 } } - 5\). Given also that the curve passes through the point (4, 20), find the equation of the curve.

4 Find \(\int _ { 2 } ^ { 5 } \left( 2 x ^ { 3 } + 3 \right) \mathrm { d } x\).

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

6 Find \(\int _ { 2 } ^ { 5 } \left( 1 - \frac { 6 } { x ^ { 3 } } \right) \mathrm { d } x\).

7 Find \(\int _ { 1 } ^ { 2 } \left( 12 x ^ { 5 } + 5 \right) \mathrm { d } x\).

9 A curve has gradient given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 \sqrt { x }\). Find the equation of the curve, given that it passes through the point \(( 9,105 )\).

10 Find \(\int _ { 1 } ^ { 2 } \left( \begin{array} { l l } x ^ { 4 } & \frac { 3 } { x ^ { 2 } } + 1 \end{array} \right) \mathrm { d } x\), showing your working.

11 Find \(\int 30 x ^ { \frac { 3 } { 2 } } \mathrm {~d} x\).

12 Find \(\int \left( x ^ { 5 } + 10 x ^ { \frac { 3 } { 2 } } \right) \mathrm { d } x\).