| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2016 |
| Session | June |
| Marks | 4 |
| Topic | Integration by Parts |
| Type | Independent multi-part (different techniques) |
| Difficulty | Moderate -0.3 Part (a) is a standard single-application integration by parts with simple limits requiring exact evaluation. Part (b) is a routine algebraic manipulation (polynomial division or rewriting as 1 - 2/(x+1)) followed by direct integration. Both are textbook exercises requiring only technique recall with no problem-solving insight, making this slightly easier than average. |
| Spec | 1.08i Integration by parts1.08j Integration using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| Obtain \(x + 1 - 2\ln | x+1 | + C\) (A0 for omission of mod signs or \(+C\)) A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Obtain \(x - 2\ln | x+1 | + C\) (A0 for omission of mod signs or \(+C\)) A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Obtain \(x - 2\ln | x+1 | + C\) A1 |
**(a)**
Use integration by parts with $f(x) = x$ and $g'(x) = e^{-x}$ **M1**
Obtain $-xe^{-x} - e^{-x}$ **A1**
Substitute limits in the correct order with subtraction. This must be seen if wrong answer obtained **M1**
Obtain $1 - \frac{2}{e}$ with no sight of decimals **A1**
**[4]**
**(b)**
Use $u = x+1$ and substitute into the given integral **M1**
Obtain $\int\frac{u-2}{u}\mathrm{d}u$ **A1**
Simplify to two terms and integrate or use by parts if integrating $u^{-1}$ and differentiating $(u-2)$ **M1**
Obtain $x + 1 - 2\ln|x+1| + C$ (A0 for omission of mod signs or $+C$) **A1**
**[4]**
*Alternative method 1:*
Obtain $1 + \frac{k}{x+1}$ **M1**
Obtain $1 - \frac{2}{x+1}$ **A1**
Attempt to integrate to obtain $x + k\ln(x+1)$ **M1**
Obtain $x - 2\ln|x+1| + C$ (A0 for omission of mod signs or $+C$) **A1**
*Alternative method 2:*
Use parts on $(x-1)(x+1)^{-1}$ and obtain $(x-1)\ln(x+1)$ with a valid attempt at $\int\ln(x+1)\mathrm{d}x$ **M1**
Find $\int\ln(x+1)\mathrm{d}x$, dealing with $\int\frac{x}{x+1}\mathrm{d}x$ **M1**
Obtain $(x-1)\ln(x+1) - (x+1)\ln(x+1) + (x+1)$ **A1**
Obtain $x - 2\ln|x+1| + C$ **A1**8
\begin{enumerate}[label=(\alph*)]
\item Evaluate exactly $\int _ { 0 } ^ { 1 } x \mathrm { e } ^ { - x } \mathrm {~d} x$.
\item Find $\int \frac { x - 1 } { x + 1 } \mathrm {~d} x$.
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2016 Q8 [4]}}