| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2005 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indefinite & Definite Integrals |
| Type | Integration with algebraic manipulation |
| Difficulty | Moderate -0.8 This is a straightforward C2 integration question with two routine parts: (i) requires expanding brackets then integrating a polynomial (standard technique), and (ii) is direct integration of x^(-1/2) with simple limits. Both parts are textbook exercises requiring only recall of basic integration rules with minimal problem-solving. |
| Spec | 1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\int(2x^2 + 7x + 3) \, dx\) | M1 | For expanding and integration attempt |
| \(= \frac{2}{3}x^3 + \frac{7}{2}x^2 + 3x + c\) | A1, A1, B1 | For at least one term correct; For all three terms correct; For addition of arbitrary constant, and no \(\int\) or \(dx\) |
| (ii) \(\int 2x \, dx = 6\) | M1, M1, A1 | For integral of the form \(kx^3\); For evaluating at least \(F(9)\), following attempt at integration; For final answer of 6 only |
**(i)** $\int(2x^2 + 7x + 3) \, dx$ | M1 | For expanding and integration attempt
$= \frac{2}{3}x^3 + \frac{7}{2}x^2 + 3x + c$ | A1, A1, B1 | For at least one term correct; For all three terms correct; For addition of arbitrary constant, and no $\int$ or $dx$
**(ii)** $\int 2x \, dx = 6$ | M1, M1, A1 | For integral of the form $kx^3$; For evaluating at least $F(9)$, following attempt at integration; For final answer of 6 only
---3 (i) Find $\int ( 2 x + 1 ) ( x + 3 ) \mathrm { d } x$.\\
(ii) Evaluate $\int _ { 0 } ^ { 9 } \frac { 1 } { \sqrt { x } } \mathrm {~d} x$.
\hfill \mbox{\textit{OCR C2 2005 Q3 [7]}}