| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2012 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Show definite integral equals specific value (algebraic/exponential substitution) |
| Difficulty | Moderate -0.3 Part (a) is a straightforward substitution with u = 6x + 1, requiring basic manipulation and evaluation at limits. Part (b) requires expanding the bracket first, then integrating standard exponential and polynomial terms. Both are routine C3 techniques with no conceptual challenges, making this slightly easier than average. |
| Spec | 1.08d Evaluate definite integrals: between limits1.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Obtain integral of form \(k(6x+1)^{\frac{1}{2}}\) | *M1 | Any constant \(k\) |
| Obtain \(6(6x+1)^{\frac{1}{2}}\) | A1 | Or (unsimplified) equiv |
| Substitute both limits and subtract | M1 | dep *M |
| Obtain \(30 - 6\) and hence 24 | A1 | AG; necessary detail needed |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Attempt expansion of integrand | M1 | To obtain (at least) 3 terms |
| Integrate \(e^{kx}\) to obtain \(\frac{1}{k}e^{kx}\) | M1 | For any constant \(k\) other than 1 |
| Obtain \(\frac{1}{2}e^{2x} + 4e^x + 4x\) | A1 | Allow \(+ c\) at this stage |
| Obtain \(\frac{1}{2}e^2 + 4e - \frac{1}{2}\) | A1 | Or equiv in terms of \(e\) simplified to three terms; no \(+ c\) now |
# Question 4(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Obtain integral of form $k(6x+1)^{\frac{1}{2}}$ | *M1 | Any constant $k$ |
| Obtain $6(6x+1)^{\frac{1}{2}}$ | A1 | Or (unsimplified) equiv |
| Substitute both limits and subtract | M1 | dep *M |
| Obtain $30 - 6$ and hence 24 | A1 | AG; necessary detail needed |
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# Question 4(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Attempt expansion of integrand | M1 | To obtain (at least) 3 terms |
| Integrate $e^{kx}$ to obtain $\frac{1}{k}e^{kx}$ | M1 | For any constant $k$ other than 1 |
| Obtain $\frac{1}{2}e^{2x} + 4e^x + 4x$ | A1 | Allow $+ c$ at this stage |
| Obtain $\frac{1}{2}e^2 + 4e - \frac{1}{2}$ | A1 | Or equiv in terms of $e$ simplified to three terms; no $+ c$ now |
---4
\begin{enumerate}[label=(\alph*)]
\item Show that $\int _ { 0 } ^ { 4 } \frac { 18 } { \sqrt { 6 x + 1 } } \mathrm {~d} x = 24$.
\item Find $\int _ { 0 } ^ { 1 } \left( \mathrm { e } ^ { x } + 2 \right) ^ { 2 } \mathrm {~d} x$, giving your answer in terms of e .
\end{enumerate}
\hfill \mbox{\textit{OCR C3 2012 Q4 [8]}}