| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2016 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indefinite & Definite Integrals |
| Type | Improper integral evaluation |
| Difficulty | Standard +0.3 Part (a) requires expanding brackets and integrating polynomials—routine C2 work. Part (b)(i) involves standard integration of negative powers and substituting limits. Part (b)(ii) introduces improper integrals by taking a limit as a→∞, which is slightly beyond typical C2 but follows directly from (b)(i) with minimal additional insight. Overall slightly easier than average due to straightforward techniques. |
| Spec | 1.08b Integrate x^n: where n != -1 and sums1.08d Evaluate definite integrals: between limits4.08c Improper integrals: infinite limits or discontinuous integrands |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\int(2x^3 - 3x^2 + 4x - 6)dx\) | M1 | Expand brackets and attempt integration; must be reasonable attempt to expand brackets, resulting in at least 3 terms; integration attempt must have increase in power by 1 for at least 3 terms |
| \(= \frac{1}{2}x^4 - x^3 + 2x^2 - 6x + c\) | A1FT | Obtain at least three correct (algebraic) terms; allow unsimplified coefficients |
| A1 [3] | Obtain fully correct expression including \(+c\); coefficients must be fully simplified; A0 if integral sign or \(dx\) still present |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\left[-6x^{-1} + 2x^{-2}\right]^a_1\) | M1 | Attempt integration; integral must be of form \(k_1x^{-1} + k_2x^{-2}\) |
| \(= (-6a^{-1} + 2a^{-2}) - (-6 + 2)\) | A1 | Obtain fully correct expression; allow unsimplified coefficients; allow presence of \(+c\) |
| \(= 4 - 6a^{-1} + 2a^{-2}\) | M1 | Attempt correct use of limits; must be \(F(a) - F(1)\); correct order and subtraction |
| A1 [4] | Obtain \(4 - 6a^{-1} + 2a^{-2}\); coefficients simplified and constant terms combined; A0 if \(+c\) present; A0 if integral sign or \(dx\) still present |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(4\) | B1FT [1] | State 4, following their (i); their (b)(i) must be of form \(k + k_1a^{-1} + k_2a^{-2}\); must appreciate a limit is required; do not allow \(4 + 0\) |
# Question 5:
## Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\int(2x^3 - 3x^2 + 4x - 6)dx$ | M1 | Expand brackets and attempt integration; must be reasonable attempt to expand brackets, resulting in at least 3 terms; integration attempt must have increase in power by 1 for at least 3 terms |
| $= \frac{1}{2}x^4 - x^3 + 2x^2 - 6x + c$ | A1FT | Obtain at least three correct (algebraic) terms; allow unsimplified coefficients |
| | A1 [3] | Obtain fully correct expression including $+c$; coefficients must be fully simplified; A0 if integral sign or $dx$ still present |
## Part (b)(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\left[-6x^{-1} + 2x^{-2}\right]^a_1$ | M1 | Attempt integration; integral must be of form $k_1x^{-1} + k_2x^{-2}$ |
| $= (-6a^{-1} + 2a^{-2}) - (-6 + 2)$ | A1 | Obtain fully correct expression; allow unsimplified coefficients; allow presence of $+c$ |
| $= 4 - 6a^{-1} + 2a^{-2}$ | M1 | Attempt correct use of limits; must be $F(a) - F(1)$; correct order and subtraction |
| | A1 [4] | Obtain $4 - 6a^{-1} + 2a^{-2}$; coefficients simplified and constant terms combined; A0 if $+c$ present; A0 if integral sign or $dx$ still present |
## Part (b)(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $4$ | B1FT [1] | State 4, following their (i); their (b)(i) must be of form $k + k_1a^{-1} + k_2a^{-2}$; must appreciate a limit is required; do not allow $4 + 0$ |
---5
\begin{enumerate}[label=(\alph*)]
\item Find $\int \left( x ^ { 2 } + 2 \right) ( 2 x - 3 ) \mathrm { d } x$.
\item \begin{enumerate}[label=(\roman*)]
\item Find, in terms of $a$, the value of $\int _ { 1 } ^ { a } \left( 6 x ^ { - 2 } - 4 x ^ { - 3 } \right) \mathrm { d } x$, where $a$ is a constant greater than 1 .
\item Deduce the value of $\int _ { 1 } ^ { \infty } \left( 6 x ^ { - 2 } - 4 x ^ { - 3 } \right) \mathrm { d } x$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR C2 2016 Q5 [8]}}