| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2014 |
| Session | June |
| Marks | 11 |
| 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 where the substitution is given explicitly. Part (i) requires recognizing that du = cos x dx and evaluating e^(2u) from 0 to 1, which is routine. Part (ii) requires differentiating using the product rule and chain rule, then solving numerically—standard techniques. Slightly above average due to the exponential-trigonometric combination and two-part structure, but no novel insight required. |
| Spec | 1.07j Differentiate exponentials: e^(kx) and a^(kx)1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.07n Stationary points: find maxima, minima using derivatives1.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| Substitute for \(x\) and \(dx\) throughout using \(u = \sin x\) and \(du = \cos x\,dx\), or equivalent | M1 | |
| Obtain integrand \(e^{2u}\) | A1 | |
| Obtain indefinite integral \(\frac{1}{2}e^{2u}\) | A1 | |
| Use limits \(u = 0, u = 1\) correctly, or equivalent | M1 | |
| Obtain answer \(\frac{1}{2}(e^2 - 1)\), or exact equivalent | A1 | 5 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Use chain rule or product rule | M1 | |
| Obtain correct terms of the derivative in any form, e.g. \(2\cos x e^{2\sin x} \cos x - e^{2\sin x}\sin x\) | A1 + A1 | |
| Equate derivative to zero and obtain a quadratic equation in \(\sin x\) | M1 | |
| Solve a 3-term quadratic and obtain a value of \(x\) | M1 | |
| Obtain answer 0.896 | A1 | 6 marks |
**(i)**
Substitute for $x$ and $dx$ throughout using $u = \sin x$ and $du = \cos x\,dx$, or equivalent | M1 |
Obtain integrand $e^{2u}$ | A1 |
Obtain indefinite integral $\frac{1}{2}e^{2u}$ | A1 |
Use limits $u = 0, u = 1$ correctly, or equivalent | M1 |
Obtain answer $\frac{1}{2}(e^2 - 1)$, or exact equivalent | A1 | 5 marks
**(ii)**
Use chain rule or product rule | M1 |
Obtain correct terms of the derivative in any form, e.g. $2\cos x e^{2\sin x} \cos x - e^{2\sin x}\sin x$ | A1 + A1 |
Equate derivative to zero and obtain a quadratic equation in $\sin x$ | M1 |
Solve a 3-term quadratic and obtain a value of $x$ | M1 |
Obtain answer 0.896 | A1 | 6 marks9\\
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The diagram shows the curve $y = \mathrm { e } ^ { 2 \sin x } \cos x$ for $0 \leqslant x \leqslant \frac { 1 } { 2 } \pi$, and its maximum point $M$.\\
(i) Using the substitution $u = \sin x$, find the exact value of the area of the shaded region bounded by the curve and the axes.\\
(ii) Find the $x$-coordinate of $M$, giving your answer correct to 3 decimal places.
\hfill \mbox{\textit{CAIE P3 2014 Q9 [11]}}