| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2016 |
| Session | June |
| Marks | 10 |
| Topic | Integration by Substitution |
| Type | Indefinite integral with non-linear substitution (algebraic/exponential/logarithmic) |
| Difficulty | Challenging +1.2 This is a well-structured integration question that guides students through a substitution (which is explicitly suggested), followed by definite integral evaluation and pattern recognition. While it requires careful manipulation and understanding of how sin behaves over different intervals, the substitution is given, the steps are clearly scaffolded, and the techniques are standard A-level fare. It's moderately above average due to the multi-part reasoning and the need to distinguish cases based on parity in part (iii), but not exceptionally challenging. |
| Spec | 1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(u = \frac{1}{x}\) and \(\frac{du}{dx} = -\frac{1}{x^2}\). So \(\int \frac{\sin(\frac{1}{x})}{x^2} dx = \int -\sin u du = \cos u + c = \cos\left(\frac{1}{x}\right) + c\) | M1*, A1, M1d*, A1 [4] | Attempt to link \(du\) and \(dx\), to obtain \(kx^{-2}\). Correct integrand in terms of \(u\). Attempt integration of their \(f(u)\) – of form \(\cos u + ...\). Correct integral in terms of \(x\), including \(+c\). |
| (ii) | M1 [3] | Attempt correct use of limits in their integral from part (i). Allow M1 for muddles with fractions, such as \(\cos(\frac{1}{n})\). |
| \(\int_\frac{\pi}{2}^{2\pi} \frac{\sin(\frac{1}{x})}{x^2} dx = -2\) | A1 | Obtain \(-2\) cwo. |
| \(\int_\frac{\pi}{n}^{2\pi} \frac{\sin(\frac{1}{x})}{x^2} dx = 2\) | A1 | Obtain 2 cwo. |
| (iii) \(\int_{1/(n\pi)}^{1/\pi} \frac{\sin(\frac{1}{x})}{x^2} dx = \cos(n\pi) - \cos((n+1)\pi)\) | B1 | Correct general expression in terms of \(n\) (no FT on incorrect integral). |
| \(\cos(n\pi) = 1\) if \(n\) is even and \(-1\) if \(n\) is odd. So the integral is either \(1 + 1 = 2\) if \(n\) even or \(-1 - 1 = -2\) if \(n\) odd | M1, A1 [3] | Consider values of \(\cos(n\pi)\), or another relevant expression e.g. \(-2\sin(n\pi + \frac{n}{2})\). Fully convincing argument (including relevant subtractions) from cwo. |
**(i)** $u = \frac{1}{x}$ and $\frac{du}{dx} = -\frac{1}{x^2}$. So $\int \frac{\sin(\frac{1}{x})}{x^2} dx = \int -\sin u du = \cos u + c = \cos\left(\frac{1}{x}\right) + c$ | M1*, A1, M1d*, A1 [4] | Attempt to link $du$ and $dx$, to obtain $kx^{-2}$. Correct integrand in terms of $u$. Attempt integration of their $f(u)$ – of form $\cos u + ...$. Correct integral in terms of $x$, including $+c$.
**(ii)** | M1 [3] | Attempt correct use of limits in their integral from part (i). Allow M1 for muddles with fractions, such as $\cos(\frac{1}{n})$.
$\int_\frac{\pi}{2}^{2\pi} \frac{\sin(\frac{1}{x})}{x^2} dx = -2$ | A1 | Obtain $-2$ cwo.
$\int_\frac{\pi}{n}^{2\pi} \frac{\sin(\frac{1}{x})}{x^2} dx = 2$ | A1 | Obtain 2 cwo.
**(iii)** $\int_{1/(n\pi)}^{1/\pi} \frac{\sin(\frac{1}{x})}{x^2} dx = \cos(n\pi) - \cos((n+1)\pi)$ | B1 | Correct general expression in terms of $n$ (no FT on incorrect integral).
$\cos(n\pi) = 1$ if $n$ is even and $-1$ if $n$ is odd. So the integral is either $1 + 1 = 2$ if $n$ even or $-1 - 1 = -2$ if $n$ odd | M1, A1 [3] | Consider values of $\cos(n\pi)$, or another relevant expression e.g. $-2\sin(n\pi + \frac{n}{2})$. Fully convincing argument (including relevant subtractions) from cwo.\begin{enumerate}[label=(\roman*)]
\item Using the substitution $u = \frac{1}{x}$, or otherwise, find $\int \frac{\sin\left(\frac{1}{x}\right)}{x^2} \, dx$. [4]
\item Evaluate $\int_{\frac{1}{2\pi}}^{\frac{1}{\pi}} \frac{\sin\left(\frac{1}{x}\right)}{x^2} \, dx$ and $\int_{\frac{1}{2\pi}}^{\frac{1}{\pi}} \frac{\sin\left(\frac{1}{x}\right)}{x^2} \, dx$. [3]
\item Show that, when $n$ is a positive integer, the integral $\int_{\frac{1}{(n+1)\pi}}^{\frac{1}{n\pi}} \frac{\sin\left(\frac{1}{x}\right)}{x^2} \, dx$ takes one of the two values found in part (ii), distinguishing between the two cases. [3]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2016 Q10 [10]}}