| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2009 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Trigonometric substitution: direct evaluation |
| Difficulty | Standard +0.3 This is a straightforward application of a given substitution with standard trigonometric identities. The substitution x = tan θ is provided, leading to dx = sec² θ dθ and 1 + x² = sec² θ. The integrand simplifies to cos 2θ, which integrates easily. Limits convert cleanly (1 → π/4, √3 → π/3). While it requires careful execution of multiple steps, all techniques are standard C4 material with no novel insight needed, making it slightly easier than average. |
| Spec | 1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.08h Integration by substitution |
2 Use the substitution $x = \tan \theta$ to find the exact value of
$$\int _ { 1 } ^ { \sqrt { 3 } } \frac { 1 - x ^ { 2 } } { 1 + x ^ { 2 } } \mathrm {~d} x$$
\hfill \mbox{\textit{OCR C4 2009 Q2 [7]}}