1. 6 Integration Antiderivatives and the Rules of Integration
Integration by Substitution
Area and the Definite Integral
The Fundamental Theorem of Calculus
Evaluating Definite Integrals
Area Between Two Curves
Applications of the Definite Integral to Business and Economics
Volumes of Solids of Revolution

2. Integral (积分): Antiderivative (反导数)
In section 2.4, given position (位置), find its velocity (速度)—Derivative
Opposite problem now: given velocity, find position—Integral

3. Integral (积分): Antiderivative (反导数)
Definition: A function F is an antiderivative of f on an interval I if F (x) = f(x) for all x  I.
Example: F(x) = x3 − 2 x2 + x − 1 is an antiderivative of f(x) = 3 x2 − 4 x + 1
Remember: (xn)  = n xn−1 , (c)  = 0
Example: Let F(x) = x, G(x) = x + 2, and H(x) = x + c, (constant c). All antiderivatives of f(x) = 1.

4. Indefinite (不定的) Integral (积分)
Antidifferentiation (反微分) or Integration:
: integral sign（积分符号）
f(x) : integrand（被积函数）
c : constant of integration（积分常数）
x: independent variable（自变量）
If independent variable is t, then
Both t and x are dummy (虚拟) variables

5. Basic Integration Rules
Rule 1: , where k a constant
Check F  (x) = (kx + c)  = k + 0
Example:  dx =  x + C
Rule 2: Power Rule

6. Basic Integration Rules
Rule 3: Integral of a constant multiple of a function

7. Basic Integration Rules
Rule 4: The sum rule

8. Basic Integration Rules
Rule 5: Exponential function

9. Basic Integration Rules

10. Integration by Substitution (换元法)
Example:
Method 1：expand (2x + 4)50，then use rules
Method 2 : change of variable

11. Integration by Substitution
Detailed process:
g(x) = 2 x + 4, f( t ) = t50, f( g(x) ) = (2x + 4)50

12. Integration by Substitution (换元法)
Technique for Substitution
T1: substitute inside function
T2: substitute higher degree factor