| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2005 |
| Session | June |
| Marks | 19 |
| Paper | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Show substitution transforms integral, then apply integration by parts or further substitution |
| Difficulty | Challenging +1.8 This AEA question requires multiple sophisticated techniques: trigonometric substitution with sec θ, recognizing the derivative relationship, integration by parts on a non-standard integrand, and a final transformation to evaluate a definite integral. While each step follows established methods, the combination of techniques and the need to navigate between different forms elevates this significantly above standard A-level integration questions. |
| Spec | 1.08h Integration by substitution1.08i Integration by parts |
\begin{enumerate}
\item (a) Use the substitution $x = \sec \theta$ to show that
\end{enumerate}
$$\int \sqrt { } \left( x ^ { 2 } - 1 \right) d x$$
can be written as
$$\int \sec \theta \tan ^ { 2 } \theta \mathrm {~d} \theta$$
(3)\\
(b) Use integration by parts to show that
$$\int \sec \theta \tan ^ { 2 } \theta \mathrm {~d} \theta = \frac { 1 } { 2 } [ \sec \theta \tan \theta - \ln | \sec \theta + \tan \theta | ] + \text { constant. }$$
(c) Evaluate $\int _ { 0 } ^ { \frac { \pi } { 4 } } \sin x \sqrt { } ( \cos 2 x ) \mathrm { d } x$.
\hfill \mbox{\textit{Edexcel AEA 2005 Q7 [19]}}