| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2016 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Indefinite integral with linear substitution |
| Difficulty | Moderate -0.8 Both parts are routine integration exercises requiring standard techniques: (i) expand brackets then integrate term-by-term, (ii) direct substitution or reverse chain rule with a linear expression. These are textbook drill questions testing basic competency rather than problem-solving, making them easier than average A-level questions. |
| Spec | 1.08b Integrate x^n: where n != -1 and sums1.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| Expand to produce form \(k_1 + \frac{k_2}{x} + \frac{k_3}{x^2}\) | M1 | For non-zero constants \(k_1, k_2, k_3\); allow if middle term appears as two, so far, unsimplified terms |
| Obtain \(4x - 4 \ln x - \frac{1}{x}\) or \(4x - 4 \ln x - x^{-1}\) | A1 | Condoning absence of modulus signs but A0 if expression involves \( |
| Answer | Marks | Guidance |
|---|---|---|
| Integrate to obtain form \(k(4x + 1)^4\) | M1 | Any non-zero constant \(k\) |
| Obtain \(\frac{3}{16}(4x + 1)^4\) | A1 | With coefficient simplified |
| Include \(... + c\) or \(... + k\) at least once anywhere in answer to question 2 | B1 | Even if associated with incorrect integral |
| [5] |
## Part i
Expand to produce form $k_1 + \frac{k_2}{x} + \frac{k_3}{x^2}$ | M1 | For non-zero constants $k_1, k_2, k_3$; allow if middle term appears as two, so far, unsimplified terms
Obtain $4x - 4 \ln x - \frac{1}{x}$ or $4x - 4 \ln x - x^{-1}$ | A1 | Condoning absence of modulus signs but A0 if expression involves $|\ln x|$ or $|4 \ln x|$
## Part ii
Integrate to obtain form $k(4x + 1)^4$ | M1 | Any non-zero constant $k$
Obtain $\frac{3}{16}(4x + 1)^4$ | A1 | With coefficient simplified
Include $... + c$ or $... + k$ at least once anywhere in answer to question 2 | B1 | Even if associated with incorrect integral
| [5] |2 Find\\
(i) $\int \left( 2 - \frac { 1 } { x } \right) ^ { 2 } \mathrm {~d} x$,\\
(ii) $\int ( 4 x + 1 ) ^ { \frac { 1 } { 3 } } \mathrm {~d} x$.
\hfill \mbox{\textit{OCR C3 2016 Q2 [5]}}