| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2012 |
| Session | Specimen |
| Marks | 8 |
| Topic | Integration by Parts |
| Type | Independent multi-part (different techniques) |
| Difficulty | Moderate -0.3 Both parts are standard textbook exercises testing routine techniques: (i) requires direct integration of simple logarithmic forms, and (ii) is a classic integration by parts example with x^2 ln x. No problem-solving or novel insight required, though the multi-part structure and algebraic manipulation place it slightly below average difficulty. |
| Spec | 1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08i Integration by parts |
**(i)** Attempt integration to obtain at least one ln term **M1**
Obtain $\ln(x-2) - \ln(2x+3)$ **A1**
Obtain $\ln\dfrac{x-2}{2x+3}$ **A1**
$+ c$ **A1**
**(ii)** $u = \ln x$, $\dfrac{\mathrm{d}v}{\mathrm{d}x} = x^2$ **M1**
$\dfrac{\mathrm{d}u}{\mathrm{d}x} = \dfrac{1}{x}$, $v = \dfrac{x^3}{3}$ **M1**
Obtain an expression of the form $f(x) \pm \int g(x)\,\mathrm{d}x$ **M1**
Obtain $\dfrac{x^3 \ln x}{3} - \displaystyle\int \dfrac{x^3}{3} \times \dfrac{1}{x}\,\mathrm{d}x$ **A1**
Obtain $\dfrac{x^3 \ln x}{3} - \dfrac{x^3}{9} + (c)$ **A1**
n.b. Mark for $+c$ may be awarded in this part if withheld in **(i)**.
**Total: 8 marks**5 (i) Find $\int \left( \frac { 1 } { x - 2 } - \frac { 2 } { 2 x + 3 } \right) \mathrm { d } x$ giving your answer in its simplest form.\\
(ii) Use integration by parts to find $\int x ^ { 2 } \ln x \mathrm {~d} x$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2012 Q5 [8]}}