| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2011 |
| 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 These are straightforward integration exercises requiring direct application of standard results for exponential and reciprocal functions. Both parts involve simple linear substitutions (or recognition of chain rule reversal) with no problem-solving required—purely routine technique application typical of early C3 material. |
| Spec | 1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Obtain integral of form \(ke^{2x+1}\) | M1 | any non-zero constant \(k\) different from 6; using substitution \(u = 2x + 1\) to obtain \(ke^u\) earns M1 (but answer to be in terms of \(x\)) or equiv such as \(\frac{6}{2}e^{2x+1}\) |
| Obtain correct \(3e^{2x+1}\) | A1 | |
| Obtain integral of form \(k_1 \ln(2x + 1)\) | M1 | any non-zero constant \(k_1\); allow if brackets absent; \(k_1 \ln u\) (after sub'n) earns M1 |
| Obtain correct \(5\ln(2x + 1)\) | A1 | or equiv such as \(\frac{10}{2}\ln(2x + 1)\); condone brackets rather than modulus signs but brackets or modulus signs must be present (so that \(5\ln 2x + 1\) earns A0) |
| Include \(\cdots + c\) at least once | B1 | 5 marks available even if no marks awarded for integration |
**(i)** Obtain integral of form $ke^{2x+1}$ | M1 | any non-zero constant $k$ different from 6; using substitution $u = 2x + 1$ to obtain $ke^u$ earns M1 (but answer to be in terms of $x$) or equiv such as $\frac{6}{2}e^{2x+1}$
Obtain correct $3e^{2x+1}$ | A1 |
Obtain integral of form $k_1 \ln(2x + 1)$ | M1 | any non-zero constant $k_1$; allow if brackets absent; $k_1 \ln u$ (after sub'n) earns M1
Obtain correct $5\ln(2x + 1)$ | A1 | or equiv such as $\frac{10}{2}\ln(2x + 1)$; condone brackets rather than modulus signs but brackets or modulus signs must be present (so that $5\ln 2x + 1$ earns A0)
Include $\cdots + c$ at least once | B1 | 5 marks available even if no marks awarded for integration
---1 Find\\
(i) $\int 6 \mathrm { e } ^ { 2 x + 1 } \mathrm {~d} x$,\\
(ii) $\int 10 ( 2 x + 1 ) ^ { - 1 } \mathrm {~d} x$.
\hfill \mbox{\textit{OCR C3 2011 Q1 [5]}}