| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2010 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration using inverse trig and hyperbolic functions |
| Type | Half-angle tangent substitution t = tan(x/2) |
| Difficulty | Challenging +1.2 This is a standard FP2 half-angle substitution question with a prescribed substitution and definite integral. While it requires knowledge of the t = tan(x/2) formulas (sin x = 2t/(1+t²), dx = 2dt/(1+t²)) and careful algebraic manipulation, it's a textbook application of a well-practiced technique with no novel insight required. The algebra simplifies nicely and the limits are straightforward, making it moderately above average difficulty but routine for Further Maths students. |
| Spec | 1.08h Integration by substitution |
3 Use the substitution $t = \tan \frac { 1 } { 2 } x$ to show that
$$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \frac { 1 } { 1 - \sin x } \mathrm {~d} x = 1 + \sqrt { 3 }$$
\hfill \mbox{\textit{OCR FP2 2010 Q3 [6]}}