| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2008 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Parts |
| Type | Improper integral with parts |
| Difficulty | Standard +0.8 This is a Further Maths question requiring integration by parts with ln x, recognition of an improper integral at x=0 (where ln x → -∞), and evaluation using limits. While the integration technique is standard, the improper integral aspect and explicit limiting process elevate it above routine A-level questions but it remains a fairly standard FP3 exercise. |
| Spec | 1.08i Integration by parts4.08c Improper integrals: infinite limits or discontinuous integrands |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(\int x^3 \ln x\,dx = \dfrac{x^4}{4}\ln x - \int \dfrac{x^4}{4}\left(\dfrac{1}{x}\right)dx\) | M1 | \(...= kx^4\ln x \pm \int f(x)\), with \(f(x)\) not involving the 'original' \(\ln x\) |
| A1 | ||
| \(= \dfrac{x^4}{4}\ln x - \dfrac{x^4}{16} + c\) | A1 | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Integrand is not defined at \(x = 0\) | E1 | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(\int_0^e x^3 \ln x\,dx = \left\{\lim_{a\to 0}\int_a^e x^3\ln x\,dx\right\}\) | ||
| \(= \dfrac{3e^4}{16} - \lim_{a\to 0}\left[\dfrac{a^4}{4}\ln a - \dfrac{a^4}{16}\right]\) | M1 | \(F(e) - F(a)\) |
| But \(\lim_{a\to 0} a^4 \ln a = 0\) | B1 | Accept a general form e.g. \(\lim_{x\to 0} x^k \ln x = 0\) |
| So \(\int_0^e x^3 \ln x\,dx\) exists and \(= \dfrac{3e^4}{16}\) | A1 | 3 |
| Total | 7 |
## Question 5(a):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $\int x^3 \ln x\,dx = \dfrac{x^4}{4}\ln x - \int \dfrac{x^4}{4}\left(\dfrac{1}{x}\right)dx$ | M1 | $...= kx^4\ln x \pm \int f(x)$, with $f(x)$ not involving the 'original' $\ln x$ |
| | A1 | |
| $= \dfrac{x^4}{4}\ln x - \dfrac{x^4}{16} + c$ | A1 | 3 | Condone absence of '$+c$' |
## Question 5(b):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Integrand is not defined at $x = 0$ | E1 | 1 | OE |
## Question 5(c):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $\int_0^e x^3 \ln x\,dx = \left\{\lim_{a\to 0}\int_a^e x^3\ln x\,dx\right\}$ | | |
| $= \dfrac{3e^4}{16} - \lim_{a\to 0}\left[\dfrac{a^4}{4}\ln a - \dfrac{a^4}{16}\right]$ | M1 | $F(e) - F(a)$ |
| But $\lim_{a\to 0} a^4 \ln a = 0$ | B1 | Accept a general form e.g. $\lim_{x\to 0} x^k \ln x = 0$ |
| So $\int_0^e x^3 \ln x\,dx$ exists and $= \dfrac{3e^4}{16}$ | A1 | 3 | CSO |
| **Total** | **7** | |
---5
\begin{enumerate}[label=(\alph*)]
\item Find $\int x ^ { 3 } \ln x \mathrm {~d} x$.
\item Explain why $\int _ { 0 } ^ { \mathrm { e } } x ^ { 3 } \ln x \mathrm {~d} x$ is an improper integral.
\item Evaluate $\int _ { 0 } ^ { \mathrm { e } } x ^ { 3 } \ln x \mathrm {~d} x$, showing the limiting process used.
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2008 Q5 [7]}}