| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2013 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Parts |
| Type | Independent multi-part (different techniques) |
| Difficulty | Standard +0.3 Part (a) is a standard integration by parts application with ln(x), requiring careful arithmetic but no novel insight. Part (b) involves a straightforward substitution with cos³(4x) = cos(4x)(1-sin²(4x)), then routine integration. Both are textbook-style exercises slightly above average difficulty due to the algebraic manipulation and exact value requirements, but well within expected P3/C4 standard techniques. |
| Spec | 1.08h Integration by substitution1.08i Integration by parts |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Carry out integration by parts and reach \(ax^2\ln x + b\int\frac{1}{2}x^2\,dx\) | M1* | |
| Obtain \(2x^2\ln x - \int\frac{1}{x}\cdot 2x^2\,dx\) | A1 | |
| Obtain \(2x^2\ln x - x^2\) | A1 | |
| Use limits, having integrated twice | M1 (dep*) | |
| Confirm given result \(56\ln 2 - 12\) | A1 | [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State or imply \(\frac{du}{dx} = 4\cos 4x\) | B1 | |
| Carry out complete substitution except limits | M1 | |
| Obtain \(\int(\frac{1}{4} - \frac{1}{4}u^2)\,du\) or equivalent | A1 | |
| Integrate to obtain form \(k_1u + k_2u^3\) with non-zero constants \(k_1, k_2\) | M1 | |
| Use appropriate limits to obtain \(\frac{11}{96}\) | A1 | [5] |
## Question 8(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Carry out integration by parts and reach $ax^2\ln x + b\int\frac{1}{2}x^2\,dx$ | M1* | |
| Obtain $2x^2\ln x - \int\frac{1}{x}\cdot 2x^2\,dx$ | A1 | |
| Obtain $2x^2\ln x - x^2$ | A1 | |
| Use limits, having integrated twice | M1 (dep*) | |
| Confirm given result $56\ln 2 - 12$ | A1 | [5] |
## Question 8(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State or imply $\frac{du}{dx} = 4\cos 4x$ | B1 | |
| Carry out complete substitution except limits | M1 | |
| Obtain $\int(\frac{1}{4} - \frac{1}{4}u^2)\,du$ or equivalent | A1 | |
| Integrate to obtain form $k_1u + k_2u^3$ with non-zero constants $k_1, k_2$ | M1 | |
| Use appropriate limits to obtain $\frac{11}{96}$ | A1 | [5] |
---8
\begin{enumerate}[label=(\alph*)]
\item Show that $\int _ { 2 } ^ { 4 } 4 x \ln x \mathrm {~d} x = 56 \ln 2 - 12$.
\item Use the substitution $u = \sin 4 x$ to find the exact value of $\int _ { 0 } ^ { \frac { 1 } { 24 } \pi } \cos ^ { 3 } 4 x \mathrm {~d} x$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2013 Q8 [10]}}