Section 1 Antiderivatives and Indefinite Integrals Chapter 5 Integration Section 1 Antiderivatives and Indefinite Integrals

Learning Objectives for Section 5 Learning Objectives for Section 5.1 Antiderivatives and Indefinite Integrals The student will be able to formulate problems involving antiderivatives. The student will be able to use the formulas and properties of antiderivatives and indefinite integrals. The student will be able to solve applications using antiderivatives and indefinite integrals.

The Antiderivative Many operations in mathematics have inverses. For example, division is the inverse of multiplication. The inverse operation of finding a derivative, called the antiderivative, will now command our attention. A function F is an antiderivative of a function f if F´(x) = f (x).

Examples Find a function that has a derivative of 2x.

Examples Find a function that has a derivative of 2x. Answer: x2, since d/dx (x2) = 2x. Find a function that has a derivative of x. Answer: x2/2, since d/dx (x2/2) = x. Find a function that has a derivative of x2. Answer: x3/3, since d/dx (x3/3) = x2.

Examples (continued) Find a function that has a derivative of 2x. Answer: We already know that x2 is such a function. Other answers are x2 + 2 or x2 – 5. The above functions are all antiderivatives of 2x. Note that the antiderivative is not unique.

Uniqueness of Antiderivatives The following theorem says that antiderivatives are almost unique. Theorem 1: If a function has more than one antiderivative, then the antiderivatives differ by at most a constant.

Indefinite Integrals Let f (x) be a function. The family of all functions that are antiderivatives of f (x) is called the indefinite integral and has the symbol The symbol ∫ is called an integral sign, and the function f (x) is called the integrand. The symbol dx indicates that anti-differentiation is performed with respect to the variable x. By the previous theorem, if F(x) is any antiderivative of f, then The arbitrary constant C is called the constant of integration.

Example Find the indefinite integral of x2.

Example Find the indefinite integral of x2. Answer: , because

Indefinite Integral Formulas and Properties (power rule) It is important to note that property 4 states that a constant factor can be moved across an integral sign. A variable factor cannot be moved across an integral sign.

Examples for Power Rule ∫ 444 dx = 444x + C (power rule with n = 0) ∫ x3 dx = x4/4 + C (n = 3) ∫ 5 x-3 dx = -(5/2) x-2 + C (n = –3) ∫ x2/3 dx = (3/5) x5/3 + C (n = 2/3) ∫ (x4 + x + x1/2 + 1 + x –1/2) dx = x5/5 + x2/2 + (2/3) x3/2 + x + 2x1/2 + C But you cannot apply the power rule for n = –1: ∫ x –1 dx is not x0/0 + C (which is undefined). The integral of x –1 is the natural logarithm.

More Examples ∫ 4 ex dx = 4 ex + C ∫ 2 x-1 dx = 2 ln |x| + C

Application In spite of the prediction of a paperless computerized office, paper and paperboard production in the United States has steadily increased. In 1990 the production was 80.3 million short tons, and since 1970 production has been growing at a rate given by f ´(x) = 0.048x + 0.95, where x is years after 1970. Find f (x), and the production levels in 1970 and 2000.

Application (continued ) We need the integral of f ´(x), or Noting that f (20) = 80.3, we calculate 80.3 = (0.024)(202) + (0.95)(20) + C 80.3 = 28.6 + C C = 51.7 f (x) = 0.024 x2 + 0.95 x + 51.7

Application (continued ) The years 1970 and 2000 correspond to x = 0 and x = 30. f (0) = (0.024)(02) + (0.95)(0) + 51.7 = 51.7 f (30) = (0.024)(302) + (0.95)(30) + 51.7 = 101.8 The production was 51.7 short tons in 1970, and 101.8 short tons in 2000.