Determine the indefinite integral
∫e2tsin(3t) dt \int e^{2t} \sin(3t) \, dt ∫e2tsin(3t)dtA robotic underwater probe tracks its velocity along a direct path. Its velocity, vvv ms−1\text{ms}^{-1}ms−1, at time ttt seconds is modeled by the equation
v(t)=e2tsin(3t),t≥0 v(t) = e^{2t} \sin(3t), \quad t \geq 0 v(t)=e2tsin(3t),t≥0The probe starts at a fixed marker at t=0t = 0t=0 and travels until it first comes to a momentary halt at time T>0T > 0T>0.
Show that the total distance the probe has travelled from the marker is exactly
313(e2π3+1) metres \frac{3}{13}\left(e^{\frac{2\pi}{3}} + 1\right) \text{ metres} 133(e32π+1) metres864 exam-style questions on Edexcel A Level Maths Integration, covering 11.1 Integrating Standard Functions, 11.2 Integrating f(ax + b), 11.3 Using Trigonometric Identities, 11.4 Reverse Chain Rule, 11.5 Integration by Substitution, 11.6 Integration by Parts, 11.7 Partial Fractions, 11.8 Finding Areas, 11.9 The Trapezium Rule, 11.10 Solving Differential Equations, and 11.11 Modelling with Differential Equations. Each one has a worked solution and a mark scheme showing where the marks go.