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 AQA A Level Maths 1.11 H: Integration with exam-style questions for A Level Maths. 436 questions covering 1.11.1 Fundamental Theorem of Calculus, 1.11.2 Integrating standard functions, 1.11.3 Definite integrals and areas, 1.11.4 Integration as the limit of a sum (A-level only), 1.11.5 Integration by substitution and by parts (A-level only), 1.11.6 Integration using partial fractions (A-level only), 1.11.7 Differential equations with separable variables (A-level only), and 1.11.8 Interpreting solutions of differential equations (A-level only), matched to the AQA A Level Maths (7357) 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.
