| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2021 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Definite integral with exponentials |
| Difficulty | Moderate -0.8 Part (a) is a straightforward reverse chain rule application with exponentials requiring only recognition that the derivative of 2x+1 is 2, then substitution of limits. Part (b) requires knowing standard identities (tan²x = sec²x - 1 and cos2θ = 1-2sin²θ) but is still routine bookwork. Both parts are below-average difficulty, testing standard techniques without problem-solving. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05l Double angle formulae: and compound angle formulae1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Integrate to obtain form \(ke^{2x+1}\) | M1 | \(k \neq 12\); If \(k = 6\) need to see evidence of integration, e.g. use of square bracket notation |
| Obtain correct \(3e^{2x+1}\) | A1 | |
| Apply limits correctly to obtain \(3e^5 - 3e\) | A1 | Or exact equivalent; indices must be simplified, but allow \(e^1\). A0 for addition of \(+ c\) |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State \(\tan^2 x = \sec^2 x - 1\) | B1 | |
| Express \(4\sin^2 2x\) in the form \(k_1 + k_2 \cos 4x\) | M1 | Where \(k_1 k_2 \neq 0\) |
| Obtain correct \(2 - 2\cos 4x\) | A1 | |
| Integrate to obtain form \(k_3 \tan x + k_4 x + k_5 \sin 4x\) | M1 | Where \(k_3 k_4 k_5 \neq 0\) |
| Obtain correct \(\tan x + x - \frac{1}{2}\sin 4x\) | A1 | Condone absence of \(\ldots + c\) |
| Total | 5 |
## Question 4(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Integrate to obtain form $ke^{2x+1}$ | M1 | $k \neq 12$; If $k = 6$ need to see evidence of integration, e.g. use of square bracket notation |
| Obtain correct $3e^{2x+1}$ | A1 | |
| Apply limits correctly to obtain $3e^5 - 3e$ | A1 | Or exact equivalent; indices must be simplified, but allow $e^1$. A0 for addition of $+ c$ |
| **Total** | **3** | |
---
## Question 4(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| State $\tan^2 x = \sec^2 x - 1$ | B1 | |
| Express $4\sin^2 2x$ in the form $k_1 + k_2 \cos 4x$ | M1 | Where $k_1 k_2 \neq 0$ |
| Obtain correct $2 - 2\cos 4x$ | A1 | |
| Integrate to obtain form $k_3 \tan x + k_4 x + k_5 \sin 4x$ | M1 | Where $k_3 k_4 k_5 \neq 0$ |
| Obtain correct $\tan x + x - \frac{1}{2}\sin 4x$ | A1 | Condone absence of $\ldots + c$ |
| **Total** | **5** | |
---4
\begin{enumerate}[label=(\alph*)]
\item Find the exact value of $\int _ { 0 } ^ { 2 } 6 \mathrm { e } ^ { 2 x + 1 } \mathrm {~d} x$.
\item Find $\int \left( \tan ^ { 2 } x + 4 \sin ^ { 2 } 2 x \right) \mathrm { d } x$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2021 Q4 [8]}}