| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2006 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indefinite & Definite Integrals |
| Type | Improper integral evaluation |
| Difficulty | Standard +0.3 Part (a) is routine integration using power rule. Part (b)(i) requires standard integration and substitution of limits. Part (b)(ii) introduces improper integrals but only requires taking a limit as a→∞ of the previous result—slightly above average due to the improper integral concept, but still straightforward application of technique. |
| Spec | 1.08b Integrate x^n: where n != -1 and sums1.08d Evaluate definite integrals: between limits |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{2}{3}x^2 + 4x + c\) | M1 | For \(kx^2\) |
| A1 | For correct first term \(\frac{2}{3}x^2\), or equiv | |
| B1 | For correct second term 4x | |
| B1 | For \(+c\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int 4x^{-1}\text{d}x = [-4x^{-1}]_1^a\) | M1 | Obtain integral of the form \(kx^{-1}\) |
| M1 | Use limits \(x = a\) and \(x = 1\) | |
| A1 | Obtain \(4 - \frac{4}{a}\), or equivalent |
| Answer | Marks | Guidance |
|---|---|---|
| \(4\) | B1√ | State 4, or legitimate conclusion from their (b)(i) |
**(a)**
$\frac{2}{3}x^2 + 4x + c$ | M1 | For $kx^2$
| A1 | For correct first term $\frac{2}{3}x^2$, or equiv
| B1 | For correct second term 4x
| B1 | For $+c$
**(b)(i)**
$\int 4x^{-1}\text{d}x = [-4x^{-1}]_1^a$ | M1 | Obtain integral of the form $kx^{-1}$
| M1 | Use limits $x = a$ and $x = 1$
| A1 | Obtain $4 - \frac{4}{a}$, or equivalent
**(b)(ii)**
$4$ | B1√ | State 4, or legitimate conclusion from their (b)(i)
---6
\begin{enumerate}[label=(\alph*)]
\item Find $\int \left( x ^ { \frac { 1 } { 2 } } + 4 \right) \mathrm { d } x$.
\item \begin{enumerate}[label=(\roman*)]
\item Find the value, in terms of $a$, of $\int _ { 1 } ^ { a } 4 x ^ { - 2 } \mathrm {~d} x$, where $a$ is a constant greater than 1 .
\item Deduce the value of $\int _ { 1 } ^ { \infty } 4 x ^ { - 2 } \mathrm {~d} x$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR C2 2006 Q6 [8]}}