| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2018 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Parts |
| Type | Integration of x^n·ln(x) |
| Difficulty | Moderate -0.3 This is a straightforward integration by parts question with standard choices (u = ln x, dv = x^{-3}dx), followed by a routine definite integral evaluation. The technique is direct and the algebra is clean, making it slightly easier than average but still requiring proper execution of the method. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules1.08d Evaluate definite integrals: between limits1.08i Integration by parts |
| Answer | Marks |
|---|---|
| 3(i) | lnx 1 1 |
| Answer | Marks |
|---|---|
| x2 x x2 | M1* |
| Answer | Marks |
|---|---|
| 2 x2 x 2x2 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| equivalent | A1 | Condone without ‘+ C ’ |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
| Answer | Marks |
|---|---|
| 3(ii) | lnx b |
| Answer | Marks | Guidance |
|---|---|---|
| or equivalent | M1(dep*) | 1 1 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Obtain the given answer following full and exact working | A1 | The step ln2= 1ln4 or 2ln2 = ln4 needs to be clear. |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 3:
--- 3(i) ---
3(i) | lnx 1 1
Integrate by parts and reach a + b∫ . dx
x2 x x2 | M1*
1 lnx 1 1
Obtain ± ± ∫ . dx, or equivalent
2 x2 x 2x2 | A1
lnx 1
Complete integration correctly and obtain − − , or
2x2 4x2
equivalent | A1 | Condone without ‘+ C ’
ISW
3
Question | Answer | Marks | Guidance
--- 3(ii) ---
3(ii) | lnx b
Substitute limits correctly in an expression of the form a +
x2 x2
or equivalent | M1(dep*) | 1 1 1
− ln2− +
8 16 4
Obtain the given answer following full and exact working | A1 | The step ln2= 1ln4 or 2ln2 = ln4 needs to be clear.
2
2
Question | Answer | Marks | Guidance\begin{enumerate}[label=(\roman*)]
\item Find $\int \frac{\ln x}{x^3} \, dx$. [3]
\item Hence show that $\int_1^2 \frac{\ln x}{x^3} \, dx = \frac{1}{16}(3 - \ln 4)$. [2]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2018 Q3 [5]}}