Integration Practice Problems for AP Calculus

Review the following concepts if needed:

• Evaluating Basic Integrals for AP Calculus
• Integration by U-Substitution for AP Calculus
• Techniques of Integration for AP Calculus

Evaluate the following integrals in problems 1 to 20. No calculators are allowed. (However, you may use calculators to check your results.)

1. (x⁵ + 3x² – x + 1)dx
2. 
3. x³ (x⁴ – 10)⁵dx
4. 
5. 
6. 
7. x csc²(x²)dx
8. 
9. 
10. 
11. (e^2x )(e^4x)dx
12. 
13. ln(e^{5x + 1})dx
14. 
15. 
16. 
17. If = e^x + 2 and the point (0, 6) is on the graph of y, find y.
18. –3e^x sin(e^x )dx
19. 
20. If f (x) is the antiderivative of and f (1) = 5, find f (e).
21. 
22. 3x² sin x dx
23. 
24. 
25. 

(Calculator) indicates that calculators are permitted.

26. The graph of the velocity function of a moving particle for 0 ≤ t ≤ 10 is shown in Figure 10.6-1.



1. At what value of t is the speed of the particle the greatest?
2. At what time is the particle moving to the right?
27. Air is pumped into a spherical balloon, whose maximum radius is 10 meters. For what value of r is the rate of increase of the volume a hundred times that of the radius?
28. Evaluate
29. (Calculator) The function f is continuous and differentiable on (0, 2) with f '' (x) > 0 for all x in the interval (0, 2). Some of the points on the graph are shown below.



Which of the following is the best approximation for f '(1)?

1. f '(1) < 2
2. 0.5 < f '(1) < 1
3. 1.5 < f '(1) < 2.5
4. 2.5 < f '(1) < 3.5
5. f '(1) > 2
30. The graph of the function f '' on the interval [1, 8] is shown in Figure 10.6-2. At what value(s) of t on the open interval (1, 8), if any, does the graph of the function f ':



1. have a point of inflection?
2. have a relative maximum or minimum?
3. concave upward?
31. Evaluate
32. If the position of an object is given by x = 4 sin(πt), y = t² – 3t + 1, find the position of the object at t =2.
33. Find the slope of the tangent line to the curve r = 3 cos θ when θ = .

Solutions for these practice problems can be found at: Solutions to Integration Practice Problems for AP Calculus