In differentiation, the differential coefficient indicates that a function of is x differentiated with respect to the indicating that it is In integration, x, dx respect to x”. “with the variable of integration is shown by adding after the function to be d(the variable) . integrated. When we want to integrate a function, we use a special notation: Thus, to integrate 4x, we will write it as follows: = 2x2 + , . c c Note that along with the integral sign, there is a term of the form which must always be dx, written, and which indicates the variable involved, in our example We say that 4x is x. integrated with respect The function being integrated is called the integrand. Technically, x. integrals of this type are called indefinite integrals, to distinguish them from definite integrals, which we will deal with later. When you are required to evaluate an indefinite integral, your answer must always include a constant of integration. anti-derivative Formally, we define the as: If is a continuous function and is the f(x) F(x) function whose derivative is i.e.: , then: f(x), (x) = f(x) = + c; where c is any arbitrary constant. F(x) Activity 2 In each of the following determine the function if the derivative is given: f(x), (x) No. (x) f(x) 1. 4x 2. x4 3. 2x2 4. 0 5. 6. -2x3 7. Explain how the derivative (x) f(x)are and the function related to each other
In differentiation, the differential coefficient indicates that a function of x is differentiated with respect to x, the dx indicating that it is “with respect to x”. In integration, the variable of integration is shown by adding d(the variable) after the function to be integrated. When we want to integrate a function, we use a special notation: . Thus, to integrate 4x, we will write it as follows: = 2x2 + c , c . Note that along with the integral sign, there is a term of the form dx, which must always be written, and which indicates the variable involved, in our example x. We say that 4x is integrated with respect x. The function being integrated is called the integrand. Technically, integrals of this type are called indefinite integrals, to distinguish them from definite integrals, which we will deal with later. When you are required to evaluate an indefinite integral, your answer must always include a constant of integration. Formally, we define the anti-derivative as: If f(x) is a continuous function and F(x) is the function whose derivative is f(x), i.e.: (x) = f(x) , then: = F(x) + c; where c is any arbitrary constant. Activity 2 In each of the following determine the function f(x), if the derivative (x) is given: No. (x) f(x) 1. 4x 2. x4 3. 2x2 4. 0 5. 6. -2x3 7. Explain how the derivative (x) and the function f(x)are related to each other
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