Topic Review — Introduction to Integration
This review covers all three lessons in Introduction to Integration: anti-differentiation and indefinite integrals, integration rules for composite functions, and displacement and velocity applications.
Review Questions
- Find ∫ (5x4 − 3x² + 2) dx.
- Find ∫ (4ex − 6 cos x + 5/x) dx.
- Find ∫ (3x + 2)4 dx.
- Find ∫ e5−2x dx.
- Find ∫ sin(3x − 1) dx.
- Find ∫ 4/(1 − 2x) dx.
- Given f′(x) = 6x² − 4 sin x and f(0) = 1, find f(x).
- A particle has velocity v(t) = 6t − 12 m/s and initial position s(0) = 2 m.
(a) Find s(t). (b) Find the position when the particle is at rest. - Find ∫ [cos(x/3) − 2e−x] dx.
- The gradient of a curve is dy/dx = (2x − 1)3 and it passes through (1, 2). Find y.
- A particle has velocity v(t) = t² − 6t + 8 m/s, starting at s = 0 when t = 0.
(a) When is the particle at rest? (b) Find total distance from t = 0 to t = 5. - Explain why ∫ x−1 dx ≠ x0/0 + c, and state the correct result.
- An object moves with acceleration a(t) = 12t − 6 m/s². At t = 0, velocity is 4 m/s and position is −1 m. Find s(t) and the position when v = 0 (for t > 0).
- A ball is launched upward with velocity 20 m/s from a height of 1 m above ground. Taking upward as positive, a = −10 m/s².
(a) Find v(t) and s(t). (b) Maximum height. (c) Time when it hits the ground. - Given f′(x) = 3/(2x + 1) for x > −1/2 and f(0) = 5, find f(3). Give an exact answer.