| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2014 |
| Session | June |
| Marks | 4 |
| Topic | Integration by Substitution |
| Type | Multi-part questions combining substitution with curve/area analysis |
| Difficulty | Moderate -0.3 Part (a) is a standard substitution (u = x² + 5) leading to a straightforward logarithm evaluation—routine for A-level. Part (b) requires substitution u = x - 2 followed by expanding and integrating term-by-term, which is slightly more involved but still a textbook exercise. Overall slightly easier than average due to the direct nature of both parts. |
| Spec | 1.08h Integration by substitution |
**(a)** Attempt integration to obtain an integral in $\ln(\mathrm{f}(x))$ — M1
Substitute limits to obtain correctly $\frac{1}{2}(\ln 9 - \ln 5)$ — A1
Show clearly the use of at least one log law — M1
Obtain $\ln\frac{3}{\sqrt{5}}$ www AG — A1 **[4]**
**(b)** Attempt integration by parts with $u = x$, $\mathrm{d}u = 1$ and $\mathrm{d}v = (x-2)^{0.5}$ and $v = \frac{2}{3}(x-2)^{\frac{3}{2}}$ — M1
Obtain $kx(x-2)^{\frac{3}{2}} - m\int \mathrm{f}(x)\mathrm{d}x$ — M1
Obtain $kg(x) - m\int(x-2)^{\frac{3}{2}}\,\mathrm{d}x$ — M1
Obtain $\frac{2}{3}x(x-2)^{\frac{3}{2}} - \frac{4}{15}(x-2)^{\frac{5}{2}} + c$ — A1 **[4]**
OR
Attempt reverse substitution with $u = x-2$, $\mathrm{d}u = \mathrm{d}x$ and $\sqrt{x-2} = \sqrt{u}$ — M1
Obtain $\int(u+2)u^{0.5}\,\mathrm{d}u$ — M1
Obtain $ku^{\frac{5}{2}} + mu^{\frac{3}{2}}$ — M1
Obtain $\frac{2}{5}(x-2)^{\frac{5}{2}} + \frac{4}{3}(x-2)^{\frac{3}{2}} + c$ — A110
\begin{enumerate}[label=(\alph*)]
\item Show that $\int _ { 0 } ^ { 2 } \frac { x } { x ^ { 2 } + 5 } \mathrm {~d} x = \ln \left( \frac { 3 } { \sqrt { 5 } } \right)$.
\item Find $\int x \sqrt { x - 2 } \mathrm {~d} x$.
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2014 Q10 [4]}}