| Exam Board | Edexcel |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | Integration using inverse trig and hyperbolic functions |
| Type | Integration by parts with inverse trig |
| Difficulty | Challenging +1.2 This is a Further Maths question requiring knowledge of inverse hyperbolic functions and their derivatives, followed by a definite integral. Part (a) is straightforward application of the chain rule with arsinh derivative. Part (b) requires recognizing the connection between the derivative found and the integrand (essentially reversing the differentiation), then evaluating at limits using arsinh properties. While it requires FM-specific knowledge and careful algebraic manipulation to reach the logarithmic form, it follows a standard 'hence' structure with clear signposting and no novel problem-solving insight. |
| Spec | 1.08h Integration by substitution4.07e Inverse hyperbolic: definitions, domains, ranges4.07f Inverse hyperbolic: logarithmic forms |
4. Given that $y = \operatorname { arsinh } ( \sqrt { } x ) , x > 0$,
\begin{enumerate}[label=(\alph*)]
\item find $\frac { \mathrm { d } y } { \mathrm {~d} x }$, giving your answer as a simplified fraction.
\item Hence, or otherwise, find
$$\int _ { \frac { 1 } { 4 } } ^ { 4 } \frac { 1 } { \sqrt { [ x ( x + 1 ) ] } } \mathrm { d } x$$
giving your answer in the form $\ln \left( \frac { a + b \sqrt { } 5 } { 2 } \right)$, where $a$ and $b$ are integers.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP3 Q4 [8]}}