1. 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

2. 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.

3. Eg.
Does the sign function
Exists 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 I
Integral sign
integrand
被积表达式
Constant of integration
Variable of integration

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

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.
Solution
Suppose the equation of the curve is
Hence,
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
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

12. 基本积分表 
is const.);
Tips：
denotes

15. Eg.4 find
solution
Using formula（2）

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

17. Eg.5 Evaluate
Sol.

18. Eg.6 Evaluate
Sol.

19. Eg.7 Evaluate
Sol.

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

21. Solution
The equation of the curve is

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 producing
100 items.
Solution let f (x) be the cost function,
Then
So, the cost of producing 100 items is ￥

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