| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2007 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indefinite & Definite Integrals |
| Type | Pure definite integration |
| Difficulty | Moderate -0.8 This is a straightforward C2 integration question requiring only basic techniques: expanding brackets and integrating polynomials in part (a), and rewriting as a negative power before integrating in part (b). All steps are routine applications of standard rules with no problem-solving or insight required, making it easier than average but not trivial. |
| Spec | 1.08b Integrate x^n: where n != -1 and sums1.08d Evaluate definite integrals: between limits |
| Answer | Marks | Guidance |
|---|---|---|
| (a) (i) \(\int x^3 - 4x = \frac{1}{4}x^4 - 2x^2 + c\) | M1, A1, B1 3 | Expand and attempt integration. Obtain \(\frac{1}{4}x^4 - 2x^2\) (A0 if \(\int\) or \(dx\) still present) or \(+ c\) (mark can be given in (b) if not gained here). |
| Answer | Marks | Guidance |
|---|---|---|
| \(= 253\frac{3}{4}\) | M1, A1 2 | Use limits correctly in integration attempt (ie F(6) − F(1)). Obtain \(253\frac{3}{4}\) (answer only is M0A0). |
| (b) \(\int 6x^{-4} dx = -3x^{-2} + c\) | B1, M1, A1 3 | Use of \(\frac{1}{x^4} = x^{-3}\). Obtain integral of the form \(kx^2\). Obtain correct \(-3x^{-2}\) (\(+ c\)) (A0 if \(\int\) or \(dx\) still present, but only penalise once in question). |
(a) (i) $\int x^3 - 4x = \frac{1}{4}x^4 - 2x^2 + c$ | M1, A1, B1 3 | Expand and attempt integration. Obtain $\frac{1}{4}x^4 - 2x^2$ (A0 if $\int$ or $dx$ still present) or $+ c$ (mark can be given in (b) if not gained here).
(ii) $[\frac{1}{4}x^4 - 2x^2]_1^6$
$= (324 - 72) - (\frac{1}{4} - 2)$
$= 253\frac{3}{4}$ | M1, A1 2 | Use limits correctly in integration attempt (ie F(6) − F(1)). Obtain $253\frac{3}{4}$ (answer only is M0A0).
(b) $\int 6x^{-4} dx = -3x^{-2} + c$ | B1, M1, A1 3 | Use of $\frac{1}{x^4} = x^{-3}$. Obtain integral of the form $kx^2$. Obtain correct $-3x^{-2}$ ($+ c$) (A0 if $\int$ or $dx$ still present, but only penalise once in question).6
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Find $\int x \left( x ^ { 2 } - 4 \right) d x$
\item Hence evaluate $\int _ { 1 } ^ { 6 } x \left( x ^ { 2 } - 4 \right) d x$.
\end{enumerate}\item Find $\int \frac { 6 } { x ^ { 3 } } d x$
\end{enumerate}
\hfill \mbox{\textit{OCR C2 2007 Q6 [8]}}