Lesson 7 antidifferentiation generalized power formula-simple substitution
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This document discusses antiderivatives and indefinite integration. It defines an antiderivative as a function whose derivative is the given function. The notation for antiderivatives uses an integral sign to indicate finding a function with the given differential. Several properties of indefinite integrals are described, including the power rule and that the integral of a sum is the sum of the integrals. Examples of integrating various functions using properties like substitution are provided.
Lesson 7 antidifferentiation generalized power formula-simple substitution
- 2. OBJECTIVES • At the end of the lesson, the students are expected to: • know the relationship between differentiation and integration; • identify and explain the different parts of the integral operation; and • perform basic integration by applying the power formula and the properties of the indefinite integrals.
- 3. DEFINITION OF ANTIDERIVATIVE A function F is an antiderivative of f on an interval I if F’(x)=f(x) for all x in I. Representation of Antiderivatives If F is an antiderivative of f on an interval I, then G is an antiderivative of f on the interval I if and only if G is of the form G(x)=F(x) + C, for all x in I, where C is a constant
- 4. NOTATION FOR ANTIDERIVATIVES • Antidifferentiation (or indefinite integration) is the process of finding a function whose derivative is known and is denoted by an integral sign ∫. The general solution is denoted by 𝒇 𝒙 𝒅𝒙 = 𝑭 𝒙 + 𝑪 where: f(x) is the integrand, x is the variable of integration, F(x) is the antiderivative of f(x),
- 5. • The integral sign indicates that we are to perform the operation of integration on f(x) dx, that is, we are to find a function whose differential is f(x) dx.
- 6. GENERAL PROPERTIES OF INDEFINITE INTEGRALS • The integral of the differential of a function u is u plus an arbitrary constant C. 𝑑𝑢 = 𝑢 + 𝐶 • A constant may be written before integral sign but not a variable function. 𝑘𝑑𝑢 = 𝑘 𝑑𝑢 = 𝑘𝑢 + 𝐶
- 7. • Power formula: • If n is not equal to minus one, the integral of 𝑢 𝑛 du is obtained by adding one to the exponent and dividing by the new component. 𝑢 𝑛 𝑑𝑢 = 𝑢 𝑛+1 𝑛 + 1 + 𝐶, 𝑛 ≠ −1 • The integral of the sum of several functions is equal to the sum of the integrals of separate functions. 𝑢 + 𝑣 + ⋯ + 𝑤 𝑑𝑥 = 𝑢𝑑𝑥 + 𝑣𝑑𝑥 + ⋯ + 𝑤𝑑𝑥
- 8. THE GENERAL POWER RULE FOR INTEGRATION • If g is a differentiable function of x, then 𝑔(𝑥) 𝑛 𝑔′ 𝑥 𝑑𝑥 = 𝑔(𝑥) 𝑛+1 𝑛 + 1 + 𝐶, 𝑛 ≠ −1 Equivalently, if 𝑢 = 𝑔 𝑥 , 𝑡ℎ𝑒𝑛 𝑢 𝑛 𝑑𝑢 = 𝑢 𝑛+1 𝑛 + 1 + 𝐶, 𝑛 ≠ −1
- 9. SUBSTITUTION METHOD • Quite often, the process of integration can be simplified by use of a substitution or change of variable. The purpose of substituting a new variable is to bring the problem to a form for which the standard formula, 𝐮 𝐧 𝐝𝐮 = 𝐮 𝐧+𝟏 𝐧+𝟏 + c, n≠ 𝟏 can be applied.
- 10. EXAMPLE: Find the indefinite integral. 1. (𝑥2 + 3𝑥) 4 (2x +3)dx 2. (2𝑥2+1) 𝑑𝑥 3 2𝑥3+3𝑥+1 3. 𝑥 5 + 𝑥 𝑑𝑥 4. ln(𝑙𝑛𝑥) 𝑥𝑙𝑛𝑥 𝑑𝑥 5. ( 𝑥 + 2) 𝑥 + 1 𝑑𝑥 6. 4𝑥3 + 𝑥 4𝑥2 + 1 𝑑𝑥
- 11. 7. 𝑥3 (2 − 𝑥2 )12 𝑑𝑥 8 sin 1 2 𝛽 𝑑𝛽 (1+5 cos 1 2 𝛽) 2 9. cos 2𝑥 𝑑𝑥 1−3 sin 2𝑥 10. 𝑥+1 𝑑𝑥 (3𝑥2+6𝑥+1 )2