| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2019 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Difficulty | Moderate -0.3 Part (a) requires knowing the identity tan²θ = sec²θ - 1 before integrating, which is a standard technique but requires one conceptual step beyond direct integration. Part (b) involves algebraic manipulation (splitting the fraction) followed by straightforward integration of exponentials. Both parts are routine applications of standard techniques with minimal problem-solving required, making this slightly easier than average. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int (\sec^2 3x - 1)\, dx\) | M1 | Use of identity \(\tan^2\theta = \sec^2\theta - 1\) |
| \(= \dfrac{1}{3}\tan 3x - x + c\) | A1 A1 | A1 for \(\dfrac{1}{3}\tan 3x\), A1 for \(-x+c\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(= \int_0^1 (e^{2x} + 4e^{-x})\, dx\) | M1 | Splitting fraction correctly |
| \(= \left[\dfrac{1}{2}e^{2x} - 4e^{-x}\right]_0^1\) | A1 A1 | A1 each term |
| \(= \left(\dfrac{1}{2}e^2 - 4e^{-1}\right) - \left(\dfrac{1}{2} - 4\right)\) | M1 | Applying limits |
| \(= \dfrac{1}{2}e^2 - \dfrac{4}{e} + \dfrac{7}{2}\) | A1 | Exact value |
# Question 4(a):
$\int \tan^2 3x\, dx$
| $\int (\sec^2 3x - 1)\, dx$ | M1 | Use of identity $\tan^2\theta = \sec^2\theta - 1$ |
|---|---|---|
| $= \dfrac{1}{3}\tan 3x - x + c$ | A1 A1 | A1 for $\dfrac{1}{3}\tan 3x$, A1 for $-x+c$ |
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# Question 4(b):
$\int_0^1 \dfrac{e^{3x}+4}{e^x}\, dx$
| $= \int_0^1 (e^{2x} + 4e^{-x})\, dx$ | M1 | Splitting fraction correctly |
|---|---|---|
| $= \left[\dfrac{1}{2}e^{2x} - 4e^{-x}\right]_0^1$ | A1 A1 | A1 each term |
| $= \left(\dfrac{1}{2}e^2 - 4e^{-1}\right) - \left(\dfrac{1}{2} - 4\right)$ | M1 | Applying limits |
| $= \dfrac{1}{2}e^2 - \dfrac{4}{e} + \dfrac{7}{2}$ | A1 | Exact value |
---4
\begin{enumerate}[label=(\alph*)]
\item Find $\int \tan ^ { 2 } 3 x \mathrm {~d} x$.
\item Find the exact value of $\int _ { 0 } ^ { 1 } \frac { \mathrm { e } ^ { 3 x } + 4 } { \mathrm { e } ^ { x } } \mathrm {~d} x$. Show all necessary working.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2019 Q4 [7]}}