| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2007 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Parts |
| Type | Improper integral with parts |
| Difficulty | Challenging +1.2 This is a Further Maths question requiring integration by parts with an improper integral, but follows a highly scaffolded structure. Part (b) is standard integration by parts technique, while part (c) requires evaluating a limit as x→0⁺, which is conceptually more demanding than typical A-level but still routine for FP3 students who have practiced improper integrals. |
| Spec | 4.08c Improper integrals: infinite limits or discontinuous integrands |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| Integrand is not defined at \(x = 0\) | E1 | Total: 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(\int x^{-\frac{1}{2}} \ln x\,dx = 2x^{\frac{1}{2}}\ln x - \int 2x^{\frac{1}{2}}\left(\dfrac{1}{x}\right)dx\) | M1 | \(...= kx^{\frac{1}{2}}\ln x \pm \int f(x)\), with \(f(x)\) not involving the 'original' \(\ln x\) |
| A1 | ||
| \(= 2x^{\frac{1}{2}}\ln x - 4x^{\frac{1}{2}}(+c)\) | A1 | Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(\int_0^e \dfrac{\ln x}{\sqrt{x}}\,dx = \lim_{a \to 0}\int_a^e \dfrac{\ln x}{\sqrt{x}}\,dx\) | M1 | |
| \(= -2e^{\frac{1}{2}} - \lim_{a \to 0}\left[2a^{\frac{1}{2}}\ln a - 4a^{\frac{1}{2}}\right]\) | M1 | \(F(b) - F(a)\) |
| But \(\lim_{a \to 0} a^{\frac{1}{2}}\ln a = 0\) | B1 | Accept general form e.g. \(\lim_{x \to 0} x^k \ln x = 0\) |
| So \(\int_0^e \dfrac{\ln x}{\sqrt{x}}\,dx\) exists and \(= -2e^{\frac{1}{2}}\) | A1 | Total: 4 |
## Question 4:
### Part (a):
| Working | Marks | Guidance |
|---------|-------|----------|
| Integrand is not defined at $x = 0$ | E1 | Total: 1 | OE |
### Part (b):
| Working | Marks | Guidance |
|---------|-------|----------|
| $\int x^{-\frac{1}{2}} \ln x\,dx = 2x^{\frac{1}{2}}\ln x - \int 2x^{\frac{1}{2}}\left(\dfrac{1}{x}\right)dx$ | M1 | $...= kx^{\frac{1}{2}}\ln x \pm \int f(x)$, with $f(x)$ not involving the 'original' $\ln x$ |
| | A1 | |
| $= 2x^{\frac{1}{2}}\ln x - 4x^{\frac{1}{2}}(+c)$ | A1 | Total: 3 | Condone absence of '$+c$' |
### Part (c):
| Working | Marks | Guidance |
|---------|-------|----------|
| $\int_0^e \dfrac{\ln x}{\sqrt{x}}\,dx = \lim_{a \to 0}\int_a^e \dfrac{\ln x}{\sqrt{x}}\,dx$ | M1 | |
| $= -2e^{\frac{1}{2}} - \lim_{a \to 0}\left[2a^{\frac{1}{2}}\ln a - 4a^{\frac{1}{2}}\right]$ | M1 | $F(b) - F(a)$ |
| But $\lim_{a \to 0} a^{\frac{1}{2}}\ln a = 0$ | B1 | Accept general form e.g. $\lim_{x \to 0} x^k \ln x = 0$ |
| So $\int_0^e \dfrac{\ln x}{\sqrt{x}}\,dx$ exists and $= -2e^{\frac{1}{2}}$ | A1 | Total: 4 | |
---4
\begin{enumerate}[label=(\alph*)]
\item Explain why $\int _ { 0 } ^ { \mathrm { e } } \frac { \ln x } { \sqrt { x } } \mathrm {~d} x$ is an improper integral.\\
(1 mark)
\item Use integration by parts to find $\int x ^ { - \frac { 1 } { 2 } } \ln x \mathrm {~d} x$.\\
(3 marks)
\item Show that $\int _ { 0 } ^ { \mathrm { e } } \frac { \ln x } { \sqrt { x } } \mathrm {~d} x$ exists and find its value.\\
(4 marks)
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2007 Q4 [8]}}