| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2011 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | Integration by Parts |
| Type | Integration involving inverse trig |
| Difficulty | Standard +0.8 This is a Further Maths question requiring integration by parts with inverse trig functions, which is conceptually more demanding than standard A-level. The substitution hint in part (a) guides students, but they must recognize to set u = arcsin(x), dv = dx, then use the given result to evaluate the integral. The definite integral with specific bounds and requirement to express in exact form adds complexity beyond routine integration by parts. |
| Spec | 4.08g Derivatives: inverse trig and hyperbolic functions |
5
\begin{enumerate}[label=(\alph*)]
\item Given that $u = \sqrt { 1 - x ^ { 2 } }$, find $\frac { \mathrm { d } u } { \mathrm {~d} x }$.
\item Use integration by parts to show that
$$\int _ { 0 } ^ { \frac { \sqrt { 3 } } { 2 } } \sin ^ { - 1 } x \mathrm {~d} x = a \sqrt { 3 } \pi + b$$
where $a$ and $b$ are rational numbers.
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2011 Q5 [8]}}