| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2007 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Parts |
| Type | Integration involving inverse trig |
| Difficulty | Standard +0.3 This is a straightforward integration by parts question with inverse trig. Part (a) is routine product rule differentiation. Part (b) directly uses the result from (a) to integrate by parts with clear limits and standard simplification. While inverse trig integration is FP2 content, the question is highly scaffolded and follows a standard template, making it slightly easier than average overall. |
| Spec | 1.07q Product and quotient rules: differentiation1.08i Integration by parts |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(\frac{x}{1+x^2} + \tan^{-1}x\) | B1B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(\int_0^1 \tan^{-1}x\,dx = \left[x\tan^{-1}x\right]_0^1 - \int_0^1 \frac{x\,dx}{1+x^2}\) | M1 | either use of part (a) or integration by parts. Allow if sign error |
| \(\int \frac{x\,dx}{1+x^2} = \frac{1}{2}\ln(1+x^2)\) | M1A1F | ft on \(\int\frac{x}{1-x^2}dx\) |
| \(I = 1\cdot\tan^{-1}1 - \frac{1}{2}\ln 2\) | M1 | |
| \(= \frac{\pi}{4} - \ln\sqrt{2}\) | A1 | AG |
## Question 4:
### Part (a)
| Working | Marks | Guidance |
|---------|-------|----------|
| $\frac{x}{1+x^2} + \tan^{-1}x$ | B1B1 | |
### Part (b)
| Working | Marks | Guidance |
|---------|-------|----------|
| $\int_0^1 \tan^{-1}x\,dx = \left[x\tan^{-1}x\right]_0^1 - \int_0^1 \frac{x\,dx}{1+x^2}$ | M1 | either use of part (a) or integration by parts. Allow if sign error |
| $\int \frac{x\,dx}{1+x^2} = \frac{1}{2}\ln(1+x^2)$ | M1A1F | ft on $\int\frac{x}{1-x^2}dx$ |
| $I = 1\cdot\tan^{-1}1 - \frac{1}{2}\ln 2$ | M1 | |
| $= \frac{\pi}{4} - \ln\sqrt{2}$ | A1 | AG |
---4
\begin{enumerate}[label=(\alph*)]
\item Differentiate $x \tan ^ { - 1 } x$ with respect to $x$.
\item Show that
$$\int _ { 0 } ^ { 1 } \tan ^ { - 1 } x \mathrm {~d} x = \frac { \pi } { 4 } - \ln \sqrt { 2 }$$
(5 marks)
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2007 Q4 [7]}}