Ch5 Indefinite Integral Calculus 5.1 Antiderivatives and indefinite integral Concepts of antiderivatives and indefinite integral Brief table of indefinite integrals The property of indefinite integral

Concepts of antiderivatives and indefinite integral Def: A function F is called an antiderivative of f on an interval I if for all x in I. Eg. Sinx is an antiderivatives of cosx.

Eg. Does the sign function Exists its antiderivative on ？why ？

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.

Questions: (1) Is there only one antiderivative? (2) If not, is there any relations? Eg. （ is constant）

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）

Definition ： The family of all antiderivatives of f on the interval I is called the indefinite integral of I Integral sign integrand 被积表达式 Constant of integration Variable of integration

Eg.1 Evaluate Sol. Eva. Eg.2 solution

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. Solution Suppose the equation of the curve is Hence, And the curve passes（1，2） Therefore, the equation is

According to the definition of indefinite integral，we know Tips： The operations of Differential and Indefinite Integral are mutually inverse .

Brief table of indefinite integrals example Thinking process Getting 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

基本积分表  is const.); Tips： denotes

Eg.4 find solution Using formula（2）

Properties Sol. We have proved (1). （This is true when it is the sum of finite functions）

Eg.5 Evaluate Sol.

Eg.6 Evaluate Sol.

Eg.7 Evaluate Sol.

Eg.8 Evaluate Sol. Tips： First change the form of the integrand，then apply the formula in brief table of indefinite integrals.

Solution The equation of the curve is

Eg. 10 if the marginal cost of producing x items is 1.92-0.002x and if the cost of producing one item is ￥562, find the cost function and the cost of producing 100 items. Solution let f (x) be the cost function, Then So, the cost of producing 100 items is ￥742.081

Conclusion def. of antiderivative： Def. of indefinite integral： Brief table of indefinite integral 5.1 Mutual inverse relationship Properties for indefinite integral