| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2012 |
| Session | Specimen |
| Marks | 5 |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Reverse chain rule with linear composite |
| Difficulty | Moderate -0.8 Both parts are straightforward applications of standard techniques: (i) is a direct reverse chain rule with linear composite requiring only the adjustment factor 1/2, and (ii) uses the standard identity 1+tan²θ=sec²θ followed by immediate integration. These are routine exercises testing basic recall and simple manipulation, requiring minimal problem-solving beyond recognizing standard forms. |
| Spec | 1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08h Integration by substitution |
**(i)**
- Obtain integral of the form $k(2x+3)^5$ [M1]
- Obtain $\frac{1}{10}(2x+3)^5$ [A1]
- $+ c$ — this mark may be awarded in either (i) or (ii) [B1]
**(ii)**
- Write $1 + \tan^2 2x = \sec^2 2x$ [M1]
- Obtain $\frac{1}{2}\tan 2x$ [A1]
**Total: 5 marks**4 Find\\
(i) $\quad \int ( 2 x + 3 ) ^ { 4 } \mathrm {~d} x$\\
(ii) $\quad \int \left( 1 + \tan ^ { 2 } 2 x \right) \mathrm { d } x$
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2012 Q4 [5]}}