| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2012 |
| Session | June |
| Marks | 12 |
| Topic | Integration by Substitution |
| Type | Volume of revolution with substitution |
| Difficulty | Standard +0.3 This is a structured multi-part integration question that builds progressively. Part (i) requires a simple substitution (u = 1 + e^x) recognizable from the derivative in the numerator, followed by routine definite integration. Part (ii) uses the same substitution for a squared term, requiring more algebraic manipulation but still following standard techniques, then applies the volume of revolution formula. While it has multiple steps and requires careful algebra, all techniques are standard A-level methods with clear signposting through the 'hence' structure. Slightly above average due to the algebraic manipulation in part (ii)(a) and the multi-step nature, but no novel insights required. |
| Spec | 1.08d Evaluate definite integrals: between limits1.08h Integration by substitution4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| (i)(a) \(\ln(1 + e^x) + c\) | B1, B1 [2] | |
| (i)(b) \(\ln(1 + e^{\ln 3}) - \ln(1 + e^0) = \ln 4 - \ln 2 = \ln 2\) CAO | M1 | Use of limits |
| M1, A1 [3] | Use log rule correctly | |
| (ii)(a) Make substitution, including attempt at changing \(dx\) to \(du\); Attempt to simplify to obtain \(\int \frac{u-1}{u^2}du\) | M1, A1 | |
| \(= \int\left(\frac{1}{u} - \frac{1}{u^2}\right)du\) | M1 | Deal with integrand |
| \(= \ln(u) + \frac{1}{u} + c = \ln(1+e^x) + \frac{1}{1+e^x} + c\) CAO | \A1, A1 [5] | |
| (ii)(b) \(V = \pi\int_{\ln 3}^{\ln 3}\left(\frac{e^x}{1+e^x}\right)^2 dx\) and attempt to integrate | M1 | |
| \(= \pi\left[\ln(1+e^x) + \frac{1}{1+e^x}\right]_{\ln 3} = \pi(\ln 3 - \frac{1}{2})\) | A1 [2] | [12] |
**(i)(a)** $\ln(1 + e^x) + c$ | B1, B1 [2] |
**(i)(b)** $\ln(1 + e^{\ln 3}) - \ln(1 + e^0) = \ln 4 - \ln 2 = \ln 2$ CAO | M1 | Use of limits
| M1, A1 [3] | Use log rule correctly
**(ii)(a)** Make substitution, including attempt at changing $dx$ to $du$; Attempt to simplify to obtain $\int \frac{u-1}{u^2}du$ | M1, A1 |
$= \int\left(\frac{1}{u} - \frac{1}{u^2}\right)du$ | M1 | Deal with integrand
$= \ln(u) + \frac{1}{u} + c = \ln(1+e^x) + \frac{1}{1+e^x} + c$ CAO | \A1, A1 [5] |
**(ii)(b)** $V = \pi\int_{\ln 3}^{\ln 3}\left(\frac{e^x}{1+e^x}\right)^2 dx$ and attempt to integrate | M1 |
$= \pi\left[\ln(1+e^x) + \frac{1}{1+e^x}\right]_{\ln 3} = \pi(\ln 3 - \frac{1}{2})$ | A1 [2] | [12]\begin{enumerate}[label=(\roman*)]
\item \begin{enumerate}[label=(\alph*)]
\item Find $\int \frac{e^x}{1 + e^x} dx$. [2]
\item Hence evaluate $\int_0^{\ln 3} \frac{e^x}{1 + e^x} dx$, giving your answer in the form $\ln k$, where $k$ is an integer. [3]
\end{enumerate}
\item \begin{enumerate}[label=(\alph*)]
\item Using the substitution $u = 1 + e^x$, find $\int \left(\frac{e^x}{1 + e^x}\right)^2 dx$. [5]
\item Hence find the exact volume of the solid of revolution generated when the curve given by $y = \frac{e^x}{1 + e^x}$, between $x = -\ln 3$ and $x = \ln 3$, is rotated through $2\pi$ radians about the $x$-axis. [2]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2012 Q10 [12]}}