Year: 2022
Paper: 2
Question Number: 1
Course: LFM Pure
Section: Integration
No solution available for this problem.
Candidates appeared to be generally well prepared for most topics within the examination, but there were a few situations in questions where some did not appear to be as proficient in standard techniques as needed. In particular, the method for finding invariant lines required in question 8 and the manipulation of trigonometric functions that were needed in question 10 caused considerable difficulties for some candidates. An additional issue that occurred at numerous points in the paper relates to the direction in which a deduction is required. It is important that candidates make sure that they know which statement is the one that they should start from as they deduce the other and that it is clear in their solution that the logic has gone in the correct direction. Clarity of solution is also an issue that candidates should be aware of, especially in the situations where the result to be reached has been given. It is important to check that there are no special cases that need to be considered separately, and when dividing by a function it is necessary to confirm that the function cannot be equal to 0 (and in the case of inequalities that the function always has the same sign). When drawing diagrams and sketching graphs it is useful if significant points that need to be clear are not drawn over the lines on the page as these can be difficult to interpret during the marking process.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
\begin{questionparts}
\item By integrating one of the two terms in the integrand by parts, or otherwise, find
\[\int \left(2\sqrt{1+x^3} + \frac{3x^3}{\sqrt{1+x^3}}\right)\,\mathrm{d}x\,.\]
\item Find
\[\int (x^2+2)\frac{\sin x}{x^3}\,\mathrm{d}x\,.\]
\item
\begin{enumerate}
\item Sketch the graph with equation $y = \dfrac{\mathrm{e}^x}{x}$, giving the coordinates of any stationary points.
\item Find $a$ if
\[\int_a^{2a} \frac{\mathrm{e}^x}{x}\,\mathrm{d}x = \int_a^{2a} \frac{\mathrm{e}^x}{x^2}\,\mathrm{d}x\,.\]
\item Show that it is not possible to find distinct integers $m$ and $n$ such that
\[\int_m^n \frac{\mathrm{e}^x}{x}\,\mathrm{d}x = \int_m^n \frac{\mathrm{e}^x}{x^2}\,\mathrm{d}x\,.\]
\end{enumerate}
\end{questionparts}
Most candidates attempted this question. While a few did not recognise that the process of integration by parts applied to one of the terms would then lead to the integral of the other term appearing in the answer in such a way that they could be combined, many candidates were able to show the result in the first part clearly. Many then realised that the second part would follow from a similar process but using two applications of integration by parts. Part (iii)(a) proved relatively straightforward for many candidates and some who had struggled with the first two parts were able to successfully complete this section. Some candidates struggled to deduce the correct behaviour of the graph, in many cases assuming that the x-axis would be an asymptote on both sides of the graph. Many realised that part (iii)(b) would follow from application of integration by parts and were able to follow through the process carefully to produce a clear deduction of the required value. In part (iii)(c), while most candidates were able to identify the equation that needed to be solved, many did not notice the link with part (iii)(a), which provided the easiest explanation of why two such integers could not exist. Attempts to justify through other arguments were not often sufficiently convincing to achieve the final mark.