solve the following integration

Question

1 Evaluate ∫(2𝑥 3 − 𝑥) 4 (6𝑥 2 − 1)𝑑𝑥 (3 marks)

2. Evaluate ∫ 𝑥𝑠𝑒𝑐2 (3𝑥 2 − 1)𝑑𝑥 (3 marks)

3. A rocket is shot straight up into the air with an initial velocity of 256 feet per second. a) When does it reach its maximum height? (3 marks) b) What is its maximum height? (1 mark) c) When does it hit the ground again? (3 marks) d) What is its speed when it hits the ground? (3 marks)

4. Evaluate ∫ √4 − 𝑥 2 2 0 𝑑𝑥 (4 marks)

5. Calculate the area under the parabola 𝑦 = 𝑥 2 + 2𝑥 and above the x-axis, between 𝑥 = 0 and 𝑥 = 1 (3 marks)

6. Find the arc length of the graph of 𝑦 = 𝑥 3 2 from (1,1) to (4,8) (4 marks)

7. Find the volume of the solid generated by revolving the region under the parabola 𝑦 = 𝑥 2 above the 𝑥 − 𝑎𝑥𝑖𝑠, between 𝑥 = 0 and 𝑥 = 1; about the 𝑥 − 𝑎𝑥𝑖𝑠 (3 marks)

Answer 1

in 1 and 2, use u-substitution . 1) let u=2x³+x so du=(6x+1)dx. 2) let u=3x²-1 so du=6xdx