Year: 2009
Paper: 1
Question Number: 6
Course: LFM Pure
Section: Integration
There were significantly more candidates attempting this paper again this year (over 900 in total), and the scores were pleasing: fewer than 5% of candidates failed to get at least 20 marks, and the median mark was 48. The pure questions were the most popular as usual; about two-thirds of candidates attempted each of the pure questions, with the exceptions of question 2 (attempted by about 90%) and question 5 (attempted by about one third). The mechanics questions were only marginally more popular than the probability and statistics questions this year; about one quarter of the candidates attempted each of the mechanics questions, while the statistics questions were attempted by about one fifth of the candidates. A significant number of candidates ignored the advice on the front cover and attempted more than six questions. In general, those candidates who submitted answers to eight or more questions did fairly poorly; very few people who tackled nine or more questions gained more than 60 marks overall (as only the best six questions are taken for the final mark). This suggests that a skill lacking in many students attempting STEP is the ability to pick questions effectively. This is not required for A-levels, so must become an important part of STEP preparation. Another "rubric"-type error was failing to follow the instructions in the question. In particular, when a question says "Hence", the candidate must make (significant) use of the preceding result(s) in their answer if they wish to gain any credit. In some questions (such as question 2), many candidates gained no marks for the final part (which was worth 10 marks) as they simply quoted an answer without using any of their earlier work. There were a number of common errors which appeared across the whole paper. These included a noticeable weakness in algebraic manipulations, sometimes indicating a serious lack of understanding of the mathematics involved. As examples, one candidate tried to use the misremembered identity cos β = sin √(1 − β²), while numerous candidates made deductions of the form "if a² + b² = c², then a + b = c" at some point in their work. Fraction manipulations are also notorious in the school classroom; the effects of this weakness were felt here, too. Another common problem was a lack of direction; writing a whole page of algebraic manipulations with no sense of purpose was unlikely to either reach the requested answer or gain the candidate any marks. It is a good idea when faced with a STEP question to ask oneself, "What is the point of this (part of the) question?" or "Why has this (part of the) question been asked?" Thinking about this can be a helpful guide. One aspect of this is evidenced by pages of formulæ and equations with no explanation. It is very good practice to explain why you are doing the calculation you are, and to write sentences in English to achieve this. It also forces one to focus on the purpose of the calculations, and may help avoid some dead ends. Finally, there is a tendency among some candidates when short of time to write what they would do at this point, rather than using the limited time to actually try doing it. Such comments gain no credit; marks are only awarded for making progress in a question. STEP questions do require a greater facility with mathematics and algebraic manipulation than the A-level examinations, as well as a depth of understanding which goes beyond that expected in a typical sixth-form classroom.
Difficulty Rating: 1484.0
Difficulty Comparisons: 1
Banger Rating: 1502.7
Banger Comparisons: 2
\begin{questionparts}
\item
Show that, for $m>0\,$,
\[
\int_{1/m}^m \frac{x^2}{x+1} \, \d x
= \frac{(m-1)^3(m+1)}{2m^2}+ \ln m\,.
\]
\item
Show by means of a substitution that
\[
\int_{1/m}^m \frac1 {x^n(x+1)}\,\d x
= \int_{1/m}^m \frac {u^{n-1}}{u+1}\,\d u \,.
\]
\item
Evaluate:
\begin{itemize}
\item[\bf (a)] $\displaystyle \int_{1/2}^2 \frac {x^5+3}{x^3(x+1)}\,\d x \;;$
\item[\bf (b)] $\displaystyle \int_1^2 \frac{x^5+x^3 +1}{x^3(x+1)}\, \d x\;. $
\end{itemize}
\end{questionparts}\begin{questionparts}
\item $\,$ \begin{align*}
&& \int_{1/m}^m \frac{x^2}{x+1} \d x &= \int_{1/m}^m \left ( x- 1 + \frac{1}{x+1} \right) \d x \\
&&&= \left [ \frac{x^2}{2} - x + \ln (x+1) \right]_{1/m}^m \\
&&&= \left ( m^2/2 - m + \ln(m+1) \right)- \left ( \frac{1}{2m^2} - \frac{1}{m} + \ln\left(\frac1m+1\right) \right) \\
&&&= \frac{m^4-2m^3-1+2m}{2m^2} + \ln (m+1) - \ln(m+1) + \ln m \\
&&&= \frac{(m-1)^3(m+1)}{2m^2} + \ln m
\end{align*}
\item $\,$ \begin{align*}
u = \frac{1}x, \d x = -\frac{1}{u^2} \d u:&& \int_{1/m}^m \frac1 {x^n(x+1)}\,\d x &= \int_{u=m}^{u=1/m} \frac{1}{u^{-n}(u^{-1}+1)} \frac{-1}{u^2} \d u \\
&&&= \int_{1/m}^m \frac{u^{n-1}}{u+1} \d u
\end{align*}
\item \begin{itemize}
\item[\bf (a)] $\,$ \begin{align*}
&& I &= \int_{1/2}^2 \frac {x^5+3}{x^3(x+1)}\,\d x \\
&&&= \int_{1/2}^2 \left ( \frac{x^2}{x+1} + \frac{3}{x^3(x+1)} \right) \d x \\
&&&= \int_{1/2}^2 \frac{x^2}{x+1} \d x + 3 \int_{1/2}^2 \frac{x^2}{x+1} \d x \\
&&&= 4 \left ( \frac{(2-1)^3(2+1)}{2 \cdot 2^2} + \ln 2 \right) \\
&&&= \frac32+4 \ln 2
\end{align*}
\item[\bf (b)] $\,$ \begin{align*}
&& J &= \int_1^2 \frac{x^5+x^3 +1}{x^3(x+1)}\, \d x \\
&& K &= \int_1^2 \frac{x^5 +1}{x^3(x+1)}\, \d x\\
u = 1/x, \d x = -1/u^2 \d u: &&&= \int_{u=1}^{u=1/2} \frac{u^{-5}+1}{u^{-3}(u^{-1}+1)} \frac{-1}{u^2} \d u \\
&&&= \int_{1/2}^1 \frac{1 + u^5}{u^3(u+1)} \d u \\
\Rightarrow && K &= \frac12 \int_{1/2}^2 \frac{x^5+1}{x^3(x+1)} \d x \\
&&&= \frac{(2-1)^3(2+1)}{2 \cdot 2^2} + \ln 2 \\
&&&= \frac38 + \ln 2 \\
&& L &= \int_1^2 \frac{x^3}{x^3(x+1)} \d x \\
&&&= \ln (3) - \ln 2 \\
\Rightarrow && J &= \frac38 + \ln 3
\end{align*}
\end{itemize}
\end{questionparts}
This was found to be another hard question. While the integral in part (i) looks deceptively easy, many candidates had little idea how to approach it. Simplifying algebraic fractions via division is a standard A-level technique, and students should recognise that this is a useful approach when attempting to integrate rational functions. Others successfully used the substitution u = x + 1 to simplify the integral. One very common approach to part (i) was to use integration by parts. While this is often a useful technique, here it proves fruitless, and many candidates spent much effort without appearing to realise that they were not getting anywhere. Some thought they were making progress, but had actually misapplied the parts formula. Other candidates tried manipulating the integral in completely incorrect ways, for example by trying to use the formula for the derivative of a quotient, or the derivative of a logarithm, or stating things like ∫fg dx = ∫f ∫g dx. About one third of attempts scored zero on this question, and fewer than one third of the attempts progressed beyond part (i). The main sticking point was part (ii), where few of the candidates realised that the substitution was u = 1/x. There were two major hints in the question itself: firstly that the limits were 1/m to m, and these remained unchanged, and secondly that the 1/x^n changed into u^(n−1), which is about the same as u^n. Many candidates failed to spot this, and spent time trying other substitutions unsuccessfully. Even among those who did realise what they needed to do, many seemed unsure how to handle the limits correctly when performing a substitution. Some of part (iii) could be done even without successfully completing parts (i) and (ii). Very few candidates realised this, though, and did not even attempt part (iii) if they had got stuck earlier. Of those who did try part (iii), some used partial fractions instead of the earlier results (which was perfectly acceptable). Some of these were very successful, whereas others did not fully understand partial fractions. The most common error was writing 1/(x³(1+x)) = A/x³ + B/(1+x), which cannot possibly get the correct answer. Other candidates applied the results of parts (i) and (ii) with varying degrees of success. A common error was being careless about the limits when trying to use the result of part (ii) in the integral in (iii)(b); the limits are not of the form 1/m to m, and so more care is needed.