The document discusses various techniques for calculating integrals or antiderivatives. It explains that integration is the reverse process of differentiation and involves calculating the area under a function's curve. Some key techniques covered include: using basic integration rules for polynomials, separating fractions into partial fractions, making substitutions to transform integrals into standard forms, and applying trigonometric identities to integrals with trigonometric functions. It also discusses evaluating definite integrals between limits by calculating the antiderivative at the upper and lower bounds and taking the difference.

1. THP-FTP-UB

2.  Integration or antidifferentiation is the
reverse process of differentiation.
 The symbol 𝑓 𝑥 𝑑𝑥 denote the integral of
𝑓 𝑥 with respect to the variable 𝑥.
 For example
𝑑
𝑑𝑥
𝑥4
= 4𝑥3
, so the integral of
4𝑥3 with respect to 𝑥 is written by:
4𝑥3 𝑑𝑥 = 𝑥4

3.  See that
 Because any constant term in the original
expression becomes zero in the derivative. We
therefore acknowledge the presence of such
constant term of some value by adding a symbol
𝐶 to the result of integration:
4𝑥3 𝑑𝑥 = 𝑥4
+ 𝐶
𝑪 is called constant integration and must always be
included.

6.  Polynomial expression are integrated term by
term with the individual constant of
integration consolidated into one symbol 𝐶 to
for whole expression.
 Example

7. Integration of Functions of a Linier Function of 𝒙
If:
then:
For example:
( ) ( )  f x dx F x C
( )
( )

  
F ax b
f ax b dx C
a
7 7
6 6 (5 4)
so that (5 4)
7 7 5

    
 
x x
x dx C x dx C

9.  If the integrand is an algebraic fraction that
can be separated into its partial fractions
then each individual partial fraction can be
integrated separately.
2
1 3 2
3 2 2 1
3 2
2 1
3ln | 2| 2ln | 1|
x
dx dx
x x x x
dx dx
x x
x x C
  
  
    
 
 
    
 
 

10.  If the numerator is not of lower degree than the
denominator, the first step is to divide out.
 For example
Determine
3𝑥2+18𝑥+3
3𝑥2+5𝑥−2
𝑑𝑥 by partial fraction
First we divide 3𝑥2 + 18𝑥 + 3 by 3𝑥2 + 5𝑥 − 2, so we
get
Then, we solve 1 +
13𝑥+5
3𝑥2+5𝑥−2
𝑑𝑥 = 1𝑑𝑥 +
13𝑥+5
3𝑥2+5𝑥−2
𝑑𝑥. To solve the form
13𝑥+5
3𝑥2+5𝑥−2
𝑑𝑥 we just
can use the rule like previous example.

12. Find
Find

13. (i)
For example
(ii)
For example
( ) 1
( ) ln ( )
( ) ( )

   
f x
dx df x f x C
f x f x
 
2
2
2 2
2 3 ( 3 5)
ln 3 5
3 5 3 5
  
    
    
x d x x
dx x x C
x x x x

14. (iii)
Example
Since
1
𝑐𝑜𝑠2 𝑥
= 𝑠𝑒𝑐2
𝑥, 𝑢 = 𝑥2
,
𝑑𝑢
𝑑𝑥
= 2𝑥, 𝑠𝑜

15.  Evaluate
 © (d)

16.  The part formula is
 For example
( ) ( ) ( ) ( ) ( ) ( )  u x dv x u x v x v x du x
( ) ( )
( ) ( ) ( ) ( ) where ( ) so ( )
( ) so ( )
.

   
 
 
  
 


x
x x
x x
x x
xe dx u x dv x
u x v x v x du x u x x du x dx
dv x e dx v x e
x e e dx
xe e C

17.  Many integrals with trigonometric integrands can
be evaluated after applying trigonometric
identities.
 Trigonometric identities such as:
𝑠𝑖𝑛2 𝑥 =
1
2
1 − 𝑐𝑜𝑠2𝑥
𝑐𝑜𝑠2 𝑥 =
1
2
1 + 𝑐𝑜𝑠2𝑥
𝑠𝑖𝑛𝑥. 𝑐𝑜𝑠𝑥 =
1
2
𝑠𝑖𝑛2𝑥
 For example:
 2 1
sin 1 cos2
2
1 1
cos2
2 2
sin 2
2 4
 
 
  
 
 
xdx x dx
dx xdx
x x
C

18.  Example Then we make substitution

19.  if 𝑓integrable on 𝑎, 𝑏 , moreover 𝑎
𝑏
𝑓 𝑥 𝑑𝑥 , called
the definit integral of 𝑓 from 𝑎 to 𝑏.
Then 𝑎
𝑏
𝑓 𝑥 𝑑𝑥 = 𝐹 𝑏 − 𝐹 𝑎
which is 𝐹 be any antiderivative of 𝑓 on 𝑎, 𝑏
 For example
−1
2
2𝑥 + 3 𝑑𝑥 = 𝑥2
+ 3𝑥 −1
2
= 22
+ 3.2 − −1 2
+ 3. −1 = 10 − −2 = 12
The techniques integration of definite integrals are
same with indefinite integral.

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