| Exam Board | SPS |
|---|---|
| Module | SPS SM Statistics (SPS SM Statistics) |
| Year | 2022 |
| Session | February |
| Marks | 11 |
| Topic | Integration by Substitution |
| Type | Show definite integral equals specific value (requiring partial fractions or complex algebra) |
| Difficulty | Challenging +1.2 This is a structured two-part integration question requiring substitution followed by partial fractions. While it involves multiple techniques (substitution with trigonometric functions, partial fractions, and logarithmic integration), each step is clearly signposted and follows standard A-level procedures. The substitution is given, and part (b) explicitly directs students to use 'algebraic integration' (partial fractions). This is moderately above average difficulty due to the multi-step nature and combination of techniques, but remains a standard Further Maths exercise without requiring novel insight. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08h Integration by substitution |
8.
\begin{enumerate}[label=(\alph*)]
\item Use the substitution $u = 1 + \sin ^ { 2 } x$ to show that
$$\int _ { 0 } ^ { \frac { \pi } { 6 } } \frac { 8 \tan x } { 1 + \sin ^ { 2 } x } \mathrm {~d} x = \int _ { p } ^ { q } \frac { 4 } { u ( 2 - u ) } \mathrm { d } u$$
where $p$ and $q$ are constants to be found.
\item Hence, using algebraic integration, show that
$$\int _ { 0 } ^ { \frac { \pi } { 6 } } \frac { 8 \tan x } { 1 + \sin ^ { 2 } x } \mathrm {~d} x = \ln A$$
where $A$ is a rational number to be found.\\[0pt]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Statistics 2022 Q8 [11]}}