| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2023 |
| Session | June |
| Paper | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Show substitution transforms integral, then evaluate |
| Difficulty | Challenging +1.2 This is a Further Maths FP1 question requiring the Weierstrass (t-substitution) technique with standard identities, followed by completing the square and arctan integration. While it involves multiple steps and FP1 content, the substitution is given explicitly in part (a), and part (b) follows a well-practiced procedure. The question is harder than typical A-level Core questions due to the FP1 content, but represents a standard textbook exercise for this topic rather than requiring novel insight. |
| Spec | 1.05o Trigonometric equations: solve in given intervals1.08h Integration by substitution |
\begin{enumerate}
\item (a) Show that the substitution $t = \tan \left( \frac { x } { 2 } \right)$ transforms the integral
\end{enumerate}
$$\int \frac { 1 } { 2 \sin x - \cos x + 5 } d x$$
into the integral
$$\int \frac { 1 } { 3 t ^ { 2 } + 2 t + 2 } \mathrm {~d} t$$
(b) Hence determine
$$\int \frac { 1 } { 2 \sin x - \cos x + 5 } d x$$
\hfill \mbox{\textit{Edexcel FP1 2023 Q5}}