Moderate -0.8 This is a straightforward two-part question testing basic integration skills. Part (i) requires standard integration of polynomial terms and evaluation at limits. Part (ii) is a routine 'find the curve from gradient' problem requiring integration and using an initial condition to find the constant. Both parts involve only direct application of standard techniques with no problem-solving or insight required, making it easier than average.
6. (i) Evaluate
$$\int _ { 2 } ^ { 4 } \left( 2 - \frac { 1 } { x ^ { 2 } } \right) \mathrm { d } x$$
(ii) Given that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x ^ { 3 } + 1$$
and that \(y = 3\) when \(x = 0\), find the value of \(y\) when \(x = 2\).
6. (i) Evaluate
$$\int _ { 2 } ^ { 4 } \left( 2 - \frac { 1 } { x ^ { 2 } } \right) \mathrm { d } x$$
(ii) Given that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x ^ { 3 } + 1$$
and that $y = 3$ when $x = 0$, find the value of $y$ when $x = 2$.\\
\hfill \mbox{\textit{OCR C2 Q6 [9]}}