4.1 Antiderivatives and Indefinite Integration
Objectives: •Write the general solution of a differential equation.
•Use indefinite integral notation for antiderivatives
•Use basic integration rules to find antiderivatives
Assignment: pg. 255 #’s 2‐44 even
Exploration
For each derivative, describe the original function F. a. F '( x)  2 x
b. F '( x)  x
c. F '( x)  x 2
1
x2
1
e. F '( x)  3
x
f . F '( x)  cos x
d. F '( x) 
Definition of Antiderivative
A function F is an antiderivative of f on an interval I ifF '( x)  f ( x) for all x in I. You can represent the entire family of antiderivatives of a function by adding a constant to a known antiderivative. 2
F '( x)  2 x then f ( x)  x  C
The constant C is called the constant of integration. 1
Notation for Antiderivatives
When solving a differential equation of the form
dy
 f ( x)
dx
It is convenient to write in the equivalent differential form dy  f ( x)dx
The operation of finding all solutions of this equation is called antidifferentiation (or indefinite integration) and is denoted by an integral sign ∫.
y   f ( x)dx  F ( x)  C
Variable of integration
y   f ( x)dx  F ( x)  C
Integrand
Constant of integration
An antiderivative of f (x)
The expression ∫ f (x)dx is read as the antiderivative of f with respect to x. So, the differential dx serves to identify x as the variable of integration. The term indefinite integral is a synonym for antiderivative. Basic Integration Rules
Differentiation Formulas
d
d
sin x   cos x
C   0
dx
dx
d
d
cos x    sin x
 kx   k
dx
dx
d
d
 tan x   sec2 x
 kf ( x)  kf ( x)
dx
dx
d
d
 f ( x)  g ( x)  f ( x)  g ( x) dx sec x   sec x tan x
dx
d
d n
cot x    csc2 x
 x   nx n 1
dx
dx
d
csc x    csc x cot x
dx
2
Basic Integration Rules
Integration Formulas
 cos x dx  sin x  C
 0 dx  C
 sin x dx   cos x  C
 k dx  kx  C
 sec x dx  tan x  C
 kf ( x) dx  k  f ( x)dx
 sec x  C
  f ( x)  g ( x) dx   f ( x)dx   g ( x)dx  sec x tan x dx
x
 csc x dx   cot x  C
 x dx  n  1  C, n  1
 csc x cot x dx   csc x  C
2
n 1
2
n
Find the general solution of the differential equation. dy
 3x
dx
Examples
Original
Integral


1
x3
Rewrite
Integrate
Simplify
dx
xdx
 2sin x dx
3
 ( x  2) dx
  3x

4
 5 x 2  x  dx
x 1
dx
x
sin x
dx
2
x
 cos
4