1. Integrate 2 dx x -2 dx + 2 2. Integrate (dx. Why do we have the constant of integration? Explain in your words. 3. Integrate (* + 2x dx X 4. Find the indefinite integral f(4vx -sin(x) +10)dx 2 5. Evaluate the definite integral ((x3 + + er ) dx6. Find the anti-derivative of f(t) = 2t2 + 3t3 + 4t4 dy 7. Show that y = xe'x + 2 is a solution of the initial value problem a = (1 x)?" 8. Solve the differential equation: i? = 32t +100, 5 = 50 when t = 0 t 9. Find the derivative: difsin(\\/;)dx t 4 10. A water balloon launched from the roof of a building at time t = 0 has vertical velocity v(t) = -32t + 40 feet/ sec at time t seconds, with v > 0 corresponding to upward motion. (a). If the roof of the building is 30 feet above the ground, nd an expression for the height of the water balloon above the ground at time t. (b). What is the average velocity of the balloon between t = 1.5 and t = 3 seconds? (c). A 63-foot person is standing on the ground. How fast is the water balloon falling when it strikes the person on top of the head