Year: 2011
Paper: 2
Question Number: 6
Course: LFM Pure
Section: Integration
There were just under 1000 entries for paper II this year, almost exactly the same number as last year. After the relatively easy time candidates experienced on last year's paper, this year's questions had been toughened up significantly, with particular attention made to ensure that candidates had to be prepared to invest more thought at the start of each question – last year saw far too many attempts from the weaker brethren at little more than the first part of up to ten questions, when the idea is that they should devote 25-40 minutes on four to six complete questions in order to present work of a substantial nature. It was also the intention to toughen up the final "quarter" of questions, so that a complete, or nearly-complete, conclusion to any question represented a significant (and, hopefully, satisfying) mathematical achievement. Although such matters are always best assessed with the benefit of hindsight, our efforts in these areas seem to have proved entirely successful, with the vast majority of candidates concentrating their efforts on four to six questions, as planned. Moreover, marks really did have to be earned: only around 20 candidates managed to gain or exceed a score of 100, and only a third of the entry managed to hit the half-way mark of 60. As in previous years, the pure maths questions provided the bulk of candidates' work, with relatively few efforts to be found at the applied ones. Questions 1 and 2 were attempted by almost all candidates; 3 and 4 by around three-quarters of them; 6, 7 and 9 by around half; the remaining questions were less popular, and some received almost no "hits". Overall, the highest scoring questions (averaging over half-marks) were 1, 2 and 9, along with 13 (very few attempts, but those who braved it scored very well). This at least is indicative that candidates are being careful in exercising some degree of thought when choosing (at least the first four) 'good' questions for themselves, although finding six successful questions then turned out to be a key discriminating factor of candidates' abilities from the examining team's perspective. Each of questions 4-8, 11 & 12 were rather poorly scored on, with average scores of only 5.5 to 6.6.
Difficulty Rating: 1600.0
Difficulty Comparisons: 0
Banger Rating: 1500.7
Banger Comparisons: 2
For any given function $\f$, let
\[
I = \int [\f'(x)]^2 \,[\f(x)]^n \d x\,,
\tag{$*$}
\]
where $n$ is a positive integer.
Show that, if $\f(x)$ satisfies $\f''(x) =k \f(x)\f'(x)$ for some constant
$k$, then ($*$) can be integrated to obtain an expression
for $I$ in terms of $\f(x)$, $\f'(x)$, $k$ and $n$.
\begin{questionparts}
\item
Verify your result in the case $\f(x) = \tan x\,$.
Hence find
\[
\displaystyle \int \frac{\sin^4x}{\cos^{8}x} \, \d x\;.
\]
\item Find
\[
\displaystyle \int \sec^2x\, (\sec x + \tan x)^6\,\d x\;.
\]
\end{questionparts}If $f''(x) = kf(x)f'(x)$ then we can see
\begin{align*}
&& I &= \int [\f'(x)]^2 \,[\f(x)]^n \d x \\
&&&= \int f'(x) \cdot f'(x) [f(x)]^n \d x \\
&&&= \left[ f'(x) \cdot \frac{[f(x)]^{n+1}}{n+1} \right] - \int f''(x) \frac{[f(x)]^{n+1}}{n+1} \d x \\
&&&= \frac{1}{n+1} \left (f'(x) [f(x)]^{n+1} - \int kf'(x) [f(x)]^{n+2} \d x \right) \\
&&&= \frac{1}{n+1} \left (f'(x) [f(x)]^{n+1} - k \frac{[f(x)]^{n+3}}{n+3} \right) +C\\
&&&= \frac{[f(x)]^{n+1}}{n+1} \left ( f'(x) - \frac{k[f(x)]^2}{n+3} \right) + C
\end{align*}
\begin{questionparts}
\item If $f(x) = \tan x, f'(x) = \sec^2 x, f''(x) = 2 \sec^2 x \tan x = 2 \cdot f(x) \cdot f'(x)$, so $\tan$ satisfies the conditions for the theorem.
\begin{align*}
&& I &= \int \sec^4 x \tan^n x \d x \\
&&&= \left [\sec^2x \cdot \frac{\tan^{n+1} x}{n+1} \right] - \int 2 \sec^2 x \tan x \cdot \frac{\tan^{n+1} x}{n+1} \d x \\
&&&= \left [\sec^2x \cdot \frac{\tan^{n+1} x}{n+1} \right] - 2 \cdot \frac{\tan^{n+3} x}{(n+1)(n+3)} \\
\end{align*}
So \begin{align*}
&& I &= \int \frac{\sin^4 x}{\cos^8 x} \d x \\
&&&= \int \tan^4 x \sec^4 x \d x \\
&&&= \int [\sec^2 x]^2 [\tan x]^4 \d x \\
&&&= \frac{\tan^{5}x}{5} \left ( \sec^2 x - \frac{2 \tan^2 x}{7} \right) + C
\end{align*}
\item \begin{align*}
&& I &= \int \sec^2x\, (\sec x + \tan x)^6\,\d x \\
&&&= \int (\sec x (\sec x + \tan x))^2 \cdot (\sec x + \tan x )^4 \d x \\
&&&= \frac{(\sec x + \tan x)^5}{5} \left ( \sec x (\sec x + \tan x) - \frac{(\sec x + \tan x)^2}{7} \right) + C
\end{align*}
\end{questionparts}
This was another very popular question attracting many poor scores. There were several very serious errors on display, including the beliefs that ∫[f(x)]ⁿ dx = [f(x)]^(n+1)/(n+1) or [f(x)]^(n+1)/((n+1)f'(x)). The understanding that the original integral needed to be split as ∫f'(x)·[f(x)]ⁿ dx before attempting to integrate by parts was largely absent, with many substituting immediately for f(x)f'(x) in terms of f'(x), which really wasn't helpful at all. Those who got over this initial hurdle generally coped very favourably with the rest of the question. In (i), it was quite common for candidates to omit verifying the result for tan x.