| 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 integration by parts question with a standard form (x^n·ln(x)). Students need to recognize u=ln(x), dv=x^(-1/2)dx, then apply the formula and evaluate definite integral limits. While it requires careful algebraic manipulation and evaluation at limits involving ln(4), it follows a well-practiced technique with no conceptual surprises, making it slightly easier than average. |
| Spec | 1.08i Integration by parts |
| Answer | Marks | Guidance |
|---|---|---|
| EITHER: Integrate by parts and reach \(kx^{2}\ln x-m\left[x^{2} \cdot \frac{1}{x} dx\right]\) | 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 \(k\ne^{\frac{1}{u}}-m\int e^{2 \frac{1}{u}} du\) | M1* | |
| Obtain \(2ue^{\frac{1}{u}}-2\int e^{\frac{1}{u}} du\), or equivalent | A1 | |
| Integrate again and obtain \(2ue^{\frac{1}{u}}-4e^{\frac{1}{u}}\), 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 \(k\nu\ln u-m\int u \cdot du\) | M1* | |
| Obtain \(4u\ln u-4\int \left | du\right | \), or equivalent |
| 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 \(I=\frac{x\ln x \pm x}{√x}+k\int \frac{x\ln x \pm x}{x√x} dx\) | M1* | |
| Obtain \(I=\frac{x\ln x-x}{√x}+\frac{1}{2} I-\frac{1}{2}\int \frac{1}{√x} dx\) | A1 | |
| Integrate and obtain \(I=2√x\ln x-4√x\), 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 $kx^{2}\ln x-m\left[x^{2} \cdot \frac{1}{x} dx\right]$ | 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 $k\ne^{\frac{1}{u}}-m\int e^{2 \frac{1}{u}} du$ | M1* |
Obtain $2ue^{\frac{1}{u}}-2\int e^{\frac{1}{u}} du$, or equivalent | A1 |
Integrate again and obtain $2ue^{\frac{1}{u}}-4e^{\frac{1}{u}}$, 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 $k\nu\ln u-m\int u \cdot du$ | M1* |
Obtain $4u\ln u-4\int \left|du\right|$, 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 $I=\frac{x\ln x \pm x}{√x}+k\int \frac{x\ln x \pm x}{x√x} dx$ | M1* |
Obtain $I=\frac{x\ln x-x}{√x}+\frac{1}{2} I-\frac{1}{2}\int \frac{1}{√x} dx$ | A1 |
Integrate and obtain $I=2√x\ln x-4√x$, 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]3 Find the exact value of $\int _ { 1 } ^ { 4 } \frac { \ln x } { \sqrt { } x } \mathrm {~d} x$.
\hfill \mbox{\textit{CAIE P3 2013 Q3 [5]}}