| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2013 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Parts |
| Type | Integration of x^n·ln(x) |
| Difficulty | Standard +0.3 This is a straightforward application of integration by parts with a standard choice of u = ln x and dv = x^(-1/2) dx. The integration is routine, the limits are simple to evaluate, and the question requires no problem-solving insight beyond recognizing the technique. It's slightly easier than average because it's a single-technique, single-part question with clean arithmetic. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules1.07l Derivative of ln(x): and related functions1.08i Integration by parts |
| Answer | Marks | Guidance |
|---|---|---|
| EITHER: Integrate by parts and reach \(kx^{2}\ln x - m\int x^{2}, \frac{1}{x}dx\) | M1* | |
| Obtain \(2x^{2}\ln x - 2\int \frac{1}{x^{2}}dx\), or equivalent | A1 | |
| Integrate again and obtain \(2x^{2}\ln x - 4x^{2}\), or equivalent | A1 | |
| Substitute limits \(x=1\) and \(x=4\), having integrated twice | M1(dep*) | |
| Obtain answer \(4(\ln 4-1)\), or exact equivalent | A1 | |
| OR1: Using \(u=\ln x\), or equivalent, integrate by parts and reach \(ku e^{\frac{1}{u}} - m\int e^{2^{u}}du\) | M1* | |
| Obtain \(2ue^{2^{u}} - 2\int e^{2^{u}}du\), or equivalent | A1 | |
| Integrate again and obtain \(2ue^{\frac{1}{u}} - 4e^{2^{u}}\), or equivalent | A1 | |
| Substitute limits \(u=0\) and \(u=\ln 4\), having integrated twice | M1(dep*) | |
| Obtain answer \(4\ln 4 - 4\), or exact equivalent | A1 | |
| OR2: Using \(u=\sqrt{x}\), or equivalent, integrate and obtain \(ku\ln u - m\int u, \frac{1}{u}du\) | M1* | |
| Obtain \(4u\ln u - 4\int ldu\), or equivalent | A1 | |
| Integrate again and obtain \(4u\ln u - 4u\), or equivalent | A1 | |
| Substitute limits \(u=1\) and \(u=2\), having integrated twice or quoted \(\int \ln u\,du\) as \(u\ln u \pm u\) | M1(dep*) | |
| Obtain answer \(8\ln 2 - 4\), or exact equivalent | A1 | |
| OR3: Integrate by parts and reach \(I = \frac{x\ln x \pm x}{...} + k\int \frac{x\ln x \pm x}{...}dx\) | M1* | |
| Obtain \(I = \frac{x\ln x - x}{\sqrt{x}} + \frac{1}{2} - \frac{1}{2}\int \frac{1}{\sqrt{x}}dx\) | A1 | |
| Integrate and obtain \(I = 2\sqrt{x}\ln x - 4\sqrt{x}\), or equivalent | A1 | |
| Substitute limits \(x=1\) and \(x=4\), having integrated twice | M1(dep*) | |
| Obtain answer \(4\ln 4 - 4\), or exact equivalent | A1 | [5] |
**EITHER:** Integrate by parts and reach $kx^{2}\ln x - m\int x^{2}, \frac{1}{x}dx$ | M1* |
Obtain $2x^{2}\ln x - 2\int \frac{1}{x^{2}}dx$, or equivalent | A1 |
Integrate again and obtain $2x^{2}\ln x - 4x^{2}$, or equivalent | A1 |
Substitute limits $x=1$ and $x=4$, having integrated twice | M1(dep*) |
Obtain answer $4(\ln 4-1)$, or exact equivalent | A1 |
**OR1:** Using $u=\ln x$, or equivalent, integrate by parts and reach $ku e^{\frac{1}{u}} - m\int e^{2^{u}}du$ | M1* |
Obtain $2ue^{2^{u}} - 2\int e^{2^{u}}du$, or equivalent | A1 |
Integrate again and obtain $2ue^{\frac{1}{u}} - 4e^{2^{u}}$, or equivalent | A1 |
Substitute limits $u=0$ and $u=\ln 4$, having integrated twice | M1(dep*) |
Obtain answer $4\ln 4 - 4$, or exact equivalent | A1 |
**OR2:** Using $u=\sqrt{x}$, or equivalent, integrate and obtain $ku\ln u - m\int u, \frac{1}{u}du$ | M1* |
Obtain $4u\ln u - 4\int ldu$, or equivalent | A1 |
Integrate again and obtain $4u\ln u - 4u$, or equivalent | A1 |
Substitute limits $u=1$ and $u=2$, having integrated twice or quoted $\int \ln u\,du$ as $u\ln u \pm u$ | M1(dep*) |
Obtain answer $8\ln 2 - 4$, or exact equivalent | A1 |
**OR3:** Integrate by parts and reach $I = \frac{x\ln x \pm x}{...} + k\int \frac{x\ln x \pm x}{...}dx$ | M1* |
Obtain $I = \frac{x\ln x - x}{\sqrt{x}} + \frac{1}{2} - \frac{1}{2}\int \frac{1}{\sqrt{x}}dx$ | A1 |
Integrate and obtain $I = 2\sqrt{x}\ln x - 4\sqrt{x}$, or equivalent | A1 |
Substitute limits $x=1$ and $x=4$, having integrated twice | M1(dep*) |
Obtain answer $4\ln 4 - 4$, or exact equivalent | A1 | [5]Find the exact value of $\int_1^4 \frac{\ln x}{\sqrt{x}} dx$. [5]
\hfill \mbox{\textit{CAIE P3 2013 Q3 [5]}}