| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2012 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Trigonometric substitution: direct evaluation |
| Difficulty | Standard +0.3 Part (a) is a straightforward quotient rule application with basic trigonometric identities. Part (b) is a standard trigonometric substitution integral with given substitution, requiring careful manipulation of trigonometric identities and evaluation of limits, but follows a well-practiced technique for C3 level. The 9 marks reflect length rather than conceptual difficulty. |
| Spec | 1.07q Product and quotient rules: differentiation1.08h Integration by substitution |
6
\begin{enumerate}[label=(\alph*)]
\item Given that $x = \frac { 1 } { \sin \theta }$, use the quotient rule to show that $\frac { \mathrm { d } x } { \mathrm {~d} \theta } = - \operatorname { cosec } \theta \cot \theta$.\\
(3 marks)
\item Use the substitution $x = \operatorname { cosec } \theta$ to find $\int _ { \sqrt { 2 } } ^ { 2 } \frac { 1 } { x ^ { 2 } \sqrt { x ^ { 2 } - 1 } } \mathrm {~d} x$, giving your answer to three significant figures.\\
(9 marks)
\end{enumerate}
\hfill \mbox{\textit{AQA C3 2012 Q6 [12]}}