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) metresPractise Edexcel A Level Maths Integration with exam-style questions for A Level Maths. 437 questions 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, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
