Year: 2023
Paper: 2
Question Number: 1
Course: LFM Pure
Section: Integration
No solution available for this problem.
Many candidates were able to express their reasoning clearly and presented good solutions to the questions that they attempted. There were excellent solutions seen for all of the questions. An area where candidates struggled in several questions was in the direction of the logic that was required in a solution. Some candidates failed to appreciate that separate arguments may be needed for the "if" and "only if" parts of a question and, in some cases, candidates produced correct arguments, but for the wrong direction. In several questions it was clear that candidates who used sketches or diagrams generally performed much better that those who did not. Sketches often also helped to make the solution clearer and easier to understand. Several questions on the STEP papers ask candidates to show a given result. Candidates should be aware that there is a need to present sufficient detail in their solutions so that it is clear that the reasoning is well understood.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
\begin{questionparts}
\item Show that making the substitution $x = \frac{1}{t}$ in the integral
\[\int_a^b \frac{1}{(1+x^2)^{\frac{3}{2}}}\,\mathrm{d}x\,,\]
where $b > a > 0$, gives the integral
\[\int_{b^{-1}}^{a^{-1}} \frac{-t}{(1+t^2)^{\frac{3}{2}}}\,\mathrm{d}t\,.\]
\item Evaluate:
\begin{enumerate}
\item[(a)] $\displaystyle\int_{\frac{1}{2}}^{2} \frac{1}{(1+x^2)^{\frac{3}{2}}}\,\mathrm{d}x\,;$
\item[(b)] $\displaystyle\int_{-2}^{2} \frac{1}{(1+x^2)^{\frac{3}{2}}}\,\mathrm{d}x\,.$
\end{enumerate}
\item
\begin{enumerate}
\item[(a)] Show that
\[\int_{\frac{1}{2}}^{2} \frac{1}{(1+x^2)^2}\,\mathrm{d}x = \int_{\frac{1}{2}}^{2} \frac{x^2}{(1+x^2)^2}\,\mathrm{d}x = \frac{1}{2}\int_{\frac{1}{2}}^{2} \frac{1}{1+x^2}\,\mathrm{d}x\,,\]
and hence evaluate
\[\int_{\frac{1}{2}}^{2} \frac{1}{(1+x^2)^2}\,\mathrm{d}x\,.\]
\item[(b)] Evaluate
\[\int_{\frac{1}{2}}^{2} \frac{1-x}{x(1+x^2)^{\frac{1}{2}}}\,\mathrm{d}x\,.\]
\end{enumerate}
\end{questionparts}
The first part of this question was often completed well, although candidates should note that in questions where the result is given it is important to show enough detail in the solution. Weaker candidates failed to change the limits or did not differentiate 1/x correctly when completing the substitution. Most candidates realised that part (ii)(a) could be completed by applying the result from part (i) and were able to select the correct values for a and b. However, many did not realise that the result from part (i) was not directly applicable to part (ii)(b) and so did not gain any marks for that part, although some candidates did realise that the answer of zero could not be correct and received some credit for recognising that the function was even and so could identify the start of a correct solution. Solutions that applied the result from part (i) successfully often achieved full marks, although in some cases the way in which limits were dealt with was not sufficient. A significant number of candidates recognised that part (ii)(b) could be solved with a tan substitution and while this approach was successful, in some cases the final answer was not written in its simplest form. In part (iii)(a) many candidates recognised that the same substitution would produce the required results, but as in part (i) several cases did not produce clear enough solutions to earn all of the marks. Most candidates were able to successfully calculate the value of the integral. Many candidates did not choose a suitable substitution for part (iii)(b), but those who did generally managed to reach an appropriate form of the integral that could be compared to the original. Many then deduced the correct answer from this, but several did not recognise the significance of the new integral and then attempted other substitutions with little success.