| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2009 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Parts |
| Type | Improper integral with parts |
| Difficulty | Standard +0.8 Part (a) is a standard integration by parts exercise (difficulty ~-0.5), but part (b) requires handling an improper integral where ln(x) → -∞ as x → 0⁺, demanding proper limit notation and understanding of convergence. The combination of routine technique with non-trivial limiting process elevates this to moderately challenging. |
| Spec | 4.08c Improper integrals: infinite limits or discontinuous integrands |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(\int \ln x\,dx = x\ln x - \int x\!\left(\frac{1}{x}\right)dx\) | M1 | Integration by parts |
| \(= x\ln x - x + c\) | A1 | Total: 2; CSO AG |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(\int_0^1 \ln x\,dx = \lim_{a\to 0}\int_a^1 \ln x\,dx\) | M1 | OE |
| \(= \lim_{a\to 0}\{0 - 1 - [a\ln a - a]\}\) | M1 | \(F(1) - F(a)\) OE |
| But \(\lim_{a\to 0} a\ln a = 0\) | E1 | Accept general form e.g. \(\lim_{a\to 0} a^k \ln a = 0\) |
| So \(\int_0^1 \ln x\,dx = -1\) | A1 | Total: 4 |
## Question 4:
### Part (a):
| Working | Marks | Guidance |
|---------|-------|---------|
| $\int \ln x\,dx = x\ln x - \int x\!\left(\frac{1}{x}\right)dx$ | M1 | Integration by parts |
| $= x\ln x - x + c$ | A1 | **Total: 2**; CSO AG |
### Part (b):
| Working | Marks | Guidance |
|---------|-------|---------|
| $\int_0^1 \ln x\,dx = \lim_{a\to 0}\int_a^1 \ln x\,dx$ | M1 | OE |
| $= \lim_{a\to 0}\{0 - 1 - [a\ln a - a]\}$ | M1 | $F(1) - F(a)$ OE |
| But $\lim_{a\to 0} a\ln a = 0$ | E1 | Accept general form e.g. $\lim_{a\to 0} a^k \ln a = 0$ |
| So $\int_0^1 \ln x\,dx = -1$ | A1 | **Total: 4** |
---4
\begin{enumerate}[label=(\alph*)]
\item Use integration by parts to show that $\int \ln x \mathrm {~d} x = x \ln x - x + c$, where $c$ is an arbitrary constant.
\item Hence evaluate $\int _ { 0 } ^ { 1 } \ln x \mathrm {~d} x$, showing the limiting process used.
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2009 Q4 [6]}}