Determine the indefinite integral ∫e2tsin(3t) dt\int e^{2t} \sin(3t) \, dt∫e2tsin(3t)dt
A 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≥0v(t) = e^{2t} \sin(3t), \quad t \geq 0v(t)=e2tsin(3t),t≥0 The 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) metres
Practise Edexcel A Level Maths 11.6 Integration by Parts with exam-style questions for A Level Maths. 52 questions, 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.