Presentation on theme: "Ch5 Indefinite Integral"— Presentation transcript:
1 Ch5 Indefinite Integral Calculus5.1 Antiderivatives and indefinite integralConcepts of antiderivatives and indefinite integralBrief table of indefinite integralsThe property of indefinite integral
2 Concepts of antiderivatives and indefinite integral Def: A function F is called an antiderivative of fon an interval I if for all x in I.Eg.Sinx is an antiderivatives of cosx.
3 Eg.Does the sign functionExists its antiderivative on ？why ？
4 solution It does exist. Suppose there is an antiderivative F(x) But F(x) isn’t differential at x = 0, therefore,there is no antiderivative.Tips:Every function that has jump or removable discontinuity does have its antiderivative.
5 Questions: (1) Is there only one antiderivative? (2) If not, is there any relations?Eg.（ is constant）
6 Tips： （1）if ，for any constant ， （2）If F(x) and G(x) are the antiderivatives of f(x)then（ is constant）Solution（ is constant）
7 Definition ： The family of all antiderivatives of f on the interval I is called the indefinite integral of IIntegral signintegrand被积表达式Constant of integrationVariable of integration
9 Eg.3 if a curve passes（1，2），and the tangent slope is always twice of point of tangency’ s horizontal coordinate，find the curve’s equation.SolutionSuppose the equation of the curve isHence,And the curve passes（1，2）Therefore, the equation is
10 According to the definition of indefinite integral，we know Tips：The operations of Differential and Indefinite Integral are mutually inverse .
11 Brief table of indefinite integrals exampleThinking processGetting the formula of indefinite integrals from the formulas of differential？Because the operations between differential and indefinite integral are mutual inverse, we can get the formula of indefinite integrals from the formulas of differential.Tips
22 Eg. 10 if the marginal cost of producing x items is x and if the cost of producing one item is￥562, find the cost function and the cost of producing100 items.Solution let f (x) be the cost function,ThenSo, the cost of producing 100 items is ￥
23 Conclusion def. of antiderivative： Def. of indefinite integral： Brief table of indefinite integral 5.1Mutual inverse relationshipProperties for indefinite integral