| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2012 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Multi-part questions combining substitution with curve/area analysis |
| Difficulty | Standard +0.3 This is a straightforward integration by substitution question with a standard trigonometric integrand. Part (i) requires routine application of the given substitution u = sin 2x, leading to a simple polynomial integral. Part (ii) involves recognizing the periodic nature of the function and counting periods, which is slightly above routine but still standard for P3. The question is slightly easier than average due to the substitution being provided and the techniques being well-practiced at this level. |
| Spec | 1.08d Evaluate definite integrals: between limits1.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State or imply \(du = 2\cos 2x\,dx\) or equivalent | B1 | |
| Express integrand in terms of \(u\) and \(du\) | M1 | |
| Obtain \(\int\frac{1}{2}u^3(1-u^2)\,du\) or equivalent | A1 | |
| Integration to obtain an integral of the form \(k_1 u^4 + k_2 u^6,\; k_1, k_2 \neq 0\) | M1 | |
| Use limits \(0\) and \(1\) or (if reverting to \(x\)) \(0\) and \(\frac{1}{4}\pi\) correctly | DM1 | |
| Obtain \(\frac{1}{24}\), or equivalent | A1 | [6] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use \(40\) and upper limit from part (i) in appropriate calculation | M1 | |
| Obtain \(k = 10\) with no errors seen | A1 | [2] |
## Question 7:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State or imply $du = 2\cos 2x\,dx$ or equivalent | B1 | |
| Express integrand in terms of $u$ and $du$ | M1 | |
| Obtain $\int\frac{1}{2}u^3(1-u^2)\,du$ or equivalent | A1 | |
| Integration to obtain an integral of the form $k_1 u^4 + k_2 u^6,\; k_1, k_2 \neq 0$ | M1 | |
| Use limits $0$ and $1$ or (if reverting to $x$) $0$ and $\frac{1}{4}\pi$ correctly | DM1 | |
| Obtain $\frac{1}{24}$, or equivalent | A1 | [6] |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use $40$ and upper limit from part (i) in appropriate calculation | M1 | |
| Obtain $k = 10$ with no errors seen | A1 | [2] |
---7\\
\includegraphics[max width=\textwidth, alt={}, center]{adbef77f-e2ac-40ce-a56b-cf6776534ec1-3_543_1091_1402_529}
The diagram shows part of the curve $y = \sin ^ { 3 } 2 x \cos ^ { 3 } 2 x$. The shaded region shown is bounded by the curve and the $x$-axis and its exact area is denoted by $A$.\\
(i) Use the substitution $u = \sin 2 x$ in a suitable integral to find the value of $A$.\\
(ii) Given that $\int _ { 0 } ^ { k \pi } \left| \sin ^ { 3 } 2 x \cos ^ { 3 } 2 x \right| \mathrm { d } x = 40 A$, find the value of the constant $k$.
\hfill \mbox{\textit{CAIE P3 2012 Q7 [8]}}