Moderate -0.3 This is a straightforward C2 integration question requiring standard power rule integration, finding a constant using a boundary condition, and algebraic manipulation to match a given form. All steps are routine with no conceptual challenges, making it slightly easier than average.
5. (i) Find
$$\int \left( 8 x - \frac { 2 } { x ^ { 3 } } \right) \mathrm { d } x$$
The gradient of a curve is given by
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = 8 x - \frac { 2 } { x ^ { 3 } } , \quad x \neq 0$$
and the curve passes through the point \(( 1,1 )\).
(ii) Show that the equation of the curve can be written in the form
$$y = \left( a x + \frac { b } { x } \right) ^ { 2 }$$
where \(a\) and \(b\) are integers to be found.
5. (i) Find
$$\int \left( 8 x - \frac { 2 } { x ^ { 3 } } \right) \mathrm { d } x$$
The gradient of a curve is given by
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = 8 x - \frac { 2 } { x ^ { 3 } } , \quad x \neq 0$$
and the curve passes through the point $( 1,1 )$.\\
(ii) Show that the equation of the curve can be written in the form
$$y = \left( a x + \frac { b } { x } \right) ^ { 2 }$$
where $a$ and $b$ are integers to be found.\\
\hfill \mbox{\textit{OCR C2 Q5 [7]}}