advertisement

6. Integration 6.1: Antiderivatives Example 1. What is the revenue function R(x) if the marginal revenue is f (x) = R0 (x) = 2x − 1? Generalization: Given a function f (x), want to find a function F (x) such that F 0 (x) = f (x). We call F (x) an antiderivative of f (x). DEFINITION Antiderivative If F 0 (x) = f (x), then F (x) is called an antiderivative of f (x). Theorem 1 On (a, b), if F 0 (x) = G0 (x), then F (x) = G(x) + C, where C is any constant. 1 Example 2. Find all antiderivatives of 2x. DEFINITION The Indefinite Integral The collection of all antiderivatives of a function f (x) is called the indefinite integral and is denoted by Z f (x) dx. Given a function F (x) for which F 0 (x) = f (x), then Z f (x) dx = F (x) + C, where C is an arbitrary constant and called the constant of integration. R We call f (x) the integrand. We refer to the symbol as the integral sign, which looks like an S and stands for sum. How does an integral connect with a certain sum? We will see later. Example 3. Find Z 2x dx. 2 Rules of Integration Power Rule If n 6= −1, then Z 1 xn+1 + C. n+1 xn dx = Proof. Example 4. Using the power rule to find the following indefinite integrals. Check the answers. a) Z x9 dx. b) Z 1 √ dx. x3 3 Constant Times Function Rule For any constant c, Z Z cf (x) dx = c f (x) dx. Example 5. Using the Constant Times Function Rule to find the following indefinite integrals. Check the answers. a) Z b) 3x9 dx. Z √ 3 s ds. ln3 4 Sum or Difference Rule Z Z (f (x) ± g(x)) dx = Z f (x) dx ± g(x) dx. Example 6. Using the Sum and Difference Rule to find the following indefinite integrals. Check the answers. a) Z √ ( t + t2 ) dt. b) Z r3 − r5 π 5 dr. Indefinite Integral of Exponenial Function Z ex dx = ex + C. Example 7. Find the following indefinite integral. Check the answer. Z Indefinite Integral of x−1 = eu + ue 4 du. 1 x Z 1 dx = ln|x| + C. x Example 8. Find the following integral. Check the answer. Z 8 − dw. w 6 Summary Integration Rules Z Z cf (x) dx = c f (x) dx Z Z Z (f (x) ± g(x)) dx = f (x) dx ± g(x) dx Z 1 xn+1 + C, n 6= −1 xn dx = n+1 Z ex dx = ex + C Z 1 dx = ln|x| + C x Appplications Example 9. Given the marginal average cost 2 − 100 , find the average cost C(x). x2 7 Example 10. Given the marginal cost 5ex + 20x − 50 and the fixed cost 2000, find the cost function C(x). Example 11. Find f (x) using the following information. f 0(x) = 12x2 − 8x + 2, f (4) = 208. 8