Chapter 5 Indefinite Integrals
1.Definition
1)G(x) is called an antiderivative of f(x) on I if 
2) If G(x) is an antiderivative of f(x) on I, then the most general antiderivative of f(x) on I is G(x)+C, where C is an arbitrary constant, denoted by
2. Table of Indefinite Integrals
Example1 Find f(x) if 
Example2 Find f(x) if 
Solution
Example3
Example4
Example5
Example6
Example7
Example8
Example9
Example10
Exercises
Solution 
Chapter 5 Indefinite Integrals(continued)
The Substitution Rule(换元法)
1.1)
Example1
2) a)
Example1
Example2
b)
c)
3)
Example
Example
Example
Exercises
Solution 
3.Trigonometric functions
1)
a)
Example
b)
Example
2)
a)
Example
b)
Example
3)
Example1
Example2
Example3
Example4
Exercises
Solution
第二篇：微积分大一基础知识经典讲解
Chapter1 Functions(函数)
1.Definition 1)Afunctionf is a rule that assigns to each element x in a set A exactly one element, called f(x), in a set B.
2)The set A is called the domain(定义域) of the function.
3)The range(值域) of f is the set of all possible values of f(x) as x varies through out the domain.
2.Basic Elementary Functions(基本初等函数)
1) constant functions
f(x)=c
2) power functions
3) exponential functions
domain: R range: 
4) logarithmic functions
domain: range: R
5) trigonometric functions
f(x)=sinx f(x)=cosx f(x)=tanx f(x)=cotx f(x)=secx f(x)=cscx
6) inverse trigonometric functions
3. Definition
Given two functions f and g, the composite function(复合函数) 
is defined by
Note 
Example If 
find each function and its domain.
4.Definition An elementary function(初等函数) is constructed using combinations
(addition加, subtraction减, multiplication乘, division除) and composition
starting with basic elementary functions.
Example 
isanelementary function.
isanelementary function.
1)Polynomial(多项式) Functions
where n is a nonnegative integer.
The leading coefficient(系数) 
The degree of the polynomial is n.
In particular(特别地),
The leading coefficient 
constant function
The leading coefficient 
linear function
The leading coefficient 
quadratic(二次) function
The leading coefficient 
cubic(三次) function
2)Rational(有理) Functions
where P and Q are polynomials.
3) Root Functions
4.Piecewise Defined Functions(分段函数)
5.
6.Properties(性质)
1)Symmetry(对称性)
even function: 
in its domain.
symmetric w.r.t.(with respect to关于) the y-axis.
odd function: 
in its domain.
symmetric about the origin.
2) monotonicity(单调性)
A function f is called increasing on interval(区间) I if 
It is called decreasing on I if 
3) boundedness(有界性)
4) periodicity (周期性)
Example f(x)=sinx
Chapter 2 Limits and Continuity
1.Definition We write 
and say “f(x) approaches(tends to趋向于) L as x tends to a ”
if we can make the values of f(x) arbitrarily(任意地) close to L by taking x to be sufficiently(足够地) close to a(on either side of a) but not equal to a.
Note 
means that in finding the limit of f(x) as x tends to a, we never consider x=a.In fact, f(x) need not even be defined when x=a. The only thing that matters is how f is defined near a.
2.Limit Laws
Suppose that c is a constant and the limits
exist. Then
Note From 2), we have
3.
1)
2)
Note
4.One-Sided Limits
1)left-hand limit
Definition We write 
and say “f(x) tends to L as x tends to a from left ”
if we can make the values of f(x) arbitrarily close to L by taking x to be sufficiently close to a and x less than a.
2)right-hand limit
Definition We write 
and say “f(x) tends to L as x tends to a from right ”
if we can make the values of f(x) arbitrarily close to L by taking x to be sufficiently close to a and x greater than a.
5.Theorem
Solution
Solution
6.Infinitesimals(无穷小量) and infinities(无穷大量)
1)Definition 
We say f(x) is an infinitesimal as 
is some number or 
Example1 
is an infinitesimal as
Example2 
is an infinitesimal as
2)Theorem 
and g(x) is bounded.
Note
Example 
3)Definition 
We say f(x) is an infinity as is some number or
Example1 
is an infinity as
Example2 
is an infinity as
4)Theorem
Note 
m, n are
nonnegative integer.
Exercises
