Name _____________________________ BC Calculus

Review 6.1 – 6.2 Evaluate the indefinite integral.

1. dxx∫ 3 2 2. ∫ dxxx

1

3. ∫ dxx321 4.

( )∫ dxx 321

5. ( )∫ + dxx 23 6. ∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛++ dxxx 122

3

7. ∫++ dx

xxx 12

8. ( )( )∫ −+ dxxx 231

9. ∫ dyyy 2 10. ∫ dx 11. ( )∫ + dxxx cos3sin2 12. ( )∫ − dttt cotcsc1 13. ( )∫ − θθθ dsinsec2 14. ( )∫ + dyy 1tan 2

Find the general solution of the differential equation and check the result by differentiation.

15. 23tdtdy

= 16. 23

xdxdy

=

The graph of the derivative of a function is given. Sketch the graphs of TWO functions that have the given derivative. (There is more than one correct answer.) 17. 18. Find the equation for y, given the derivative and the indicated point on the curve.

19. 12 −= xdxdy 20. x

dxdy cos=

21. A particle moves along the x-axis at a velocity of ( ) 0,1>= t

ttv . At time t = 1, its

position is x = 4. Find the acceleration and position functions for the particle. 22. The maker of a certain automobile advertises that it takes 13 seconds to accelerate from 25 km/h to 80 km/h. Assuming constant acceleration, compute the following: a) The acceleration in meters per second per second. b) The distance the car travels during the 13 seconds.

23. A cylindrical tank with a radius of 3 ft and a height of 9 ft is full of water. There is a

hole in the bottom of the tank and water is being drained out at a rate of h18π ft³/min.

1. Find a formula for the depth of the water at any time t. 2. Find a formula for the volume of water in the tank at time t. 3. How long will it take to drain the tank?

Find the indefinite integral. 25. ( )∫ + dxxx 2sin 65 26. ( ) dxxxx∫ ++ 21 2

27. dxx

x∫

cos 28. ∫+

dxxxx

2cos1cossin

29. ∫ dxxxx 22 cossin 30. ∫ xdxx 2tan2sec

31. ( )∫ dxxx 22csc 32. ∫ dxx

xx cossin

Evaluate the definite integrals.

33. ∫ +1

045 dxx 34. ∫

6

02 2cos2sin

π

dxx

x

35. ∫2

6

2 cossin

π

πθθθ d 36. ∫

−

−1

1

21 dxxx

37. ∫2

2

4

sinπ

π

dxx

x 38. ( )∫

−+

2

2

2sin2cos

π

πdx

xx