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.)
- (x5 + 3x2 – x + 1)dx
- x3 (x4 – 10)5dx
- x csc2(x2)dx
- (e2x )(e4x)dx
- ln(e5x + 1)dx
- If = ex + 2 and the point (0, 6) is on the graph of y, find y.
- –3ex sin(ex )dx
- If f (x) is the antiderivative of and f (1) = 5, find f (e).
- 3x2 sin x dx
(Calculator) indicates that calculators are permitted.
- The graph of the velocity function of a moving particle for 0 ≤ t ≤ 10 is shown in Figure 10.6-1.
- At what value of t is the speed of the particle the greatest?
- At what time is the particle moving to the right?
- 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?
- (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.
- f '(1) < 2
- 0.5 < f '(1) < 1
- 1.5 < f '(1) < 2.5
- 2.5 < f '(1) < 3.5
- f '(1) > 2
- 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 ':
- have a point of inflection?
- have a relative maximum or minimum?
- concave upward?
- 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.
- Find the slope of the tangent line to the curve r = 3 cos θ when θ = .
Which of the following is the best approximation for f '(1)?
Solutions for these practice problems can be found at: Solutions to Integration Practice Problems for AP Calculus