| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2007 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration using inverse trig and hyperbolic functions |
| Type | Integration by parts with inverse trig |
| Difficulty | Standard +0.8 This is a Further Maths question requiring differentiation of inverse trig functions and product rule, then recognizing the result to evaluate a definite integral. Part (i) involves careful algebraic simplification with inverse trig derivatives. Part (ii) requires insight to connect the derivative back to the integral, which is non-routine. The exact value calculation adds technical demand. Moderately challenging for FM students. |
| Spec | 1.07l Derivative of ln(x): and related functions1.08d Evaluate definite integrals: between limits |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Reasonable attempt at product rule | M1 | Two terms seen |
| Derive or quote diff. of \(\cos^{-1}x\) | M1 | Allow \(+\) |
| Get \(-x^2(1-x^2)^{-1/2} + (1-x^2)^{1/2}\) | A1 | |
| Tidy to \(2(1-x^2)^{1/2}\) | A1 | cao |
| (ii) Write down integral from (i) | B1 | On any \(kv(1-x^2)\) |
| Use limits correctly | M1 | In any reasonable integral |
| Tidy to \(\frac{1}{2}\pi\) | A1 | |
| SR Reasonable sub. | B1 | Replace for new variable and attempt to integrate (ignore limits) |
| Clearly get \(\frac{1}{2}\pi\) | M1, A1 |
**(i)** Reasonable attempt at product rule | M1 | Two terms seen
Derive or quote diff. of $\cos^{-1}x$ | M1 | Allow $+$
Get $-x^2(1-x^2)^{-1/2} + (1-x^2)^{1/2}$ | A1 |
Tidy to $2(1-x^2)^{1/2}$ | A1 | cao
**(ii)** Write down integral from (i) | B1 | On any $kv(1-x^2)$
Use limits correctly | M1 | In any reasonable integral
Tidy to $\frac{1}{2}\pi$ | A1 |
SR Reasonable sub. | B1 | Replace for new variable and attempt to integrate (ignore limits)
Clearly get $\frac{1}{2}\pi$ | M1, A1 |4 (i) Given that
$$y = x \sqrt { 1 - x ^ { 2 } } - \cos ^ { - 1 } x$$
find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in a simplified form.\\
(ii) Hence, or otherwise, find the exact value of $\int _ { 0 } ^ { 1 } 2 \sqrt { 1 - x ^ { 2 } } \mathrm {~d} x$.
\hfill \mbox{\textit{OCR FP2 2007 Q4 [7]}}