Challenging +1.8 This is a Further Maths arc length question requiring parametric differentiation, simplification of expressions involving exponentials and polynomials, and integration. While the calculus is standard, the algebraic manipulation to simplify √(dx/dt)² + (dy/dt)² into an integrable form requires careful work and insight. The 7-mark allocation reflects substantial computation, placing it well above average difficulty but not at the extreme end for Further Maths.
The curve $C$ has parametric equations
$$x = \frac{1}{2}e^{2t} - \frac{1}{3}t^3 - \frac{1}{2}, \quad y = 2e^t(t-1), \quad \text{for } 0 \leqslant t \leqslant 1.$$
Find the exact length of $C$. [7]
\hfill \mbox{\textit{CAIE Further Paper 2 2024 Q3 [7]}}