This document discusses integration and integration rules. It defines integration as the reverse process of differentiation, where differentiation finds the rate of change and integration finds the family of functions whose rates of change are a given function. Several basic integration rules are summarized that allow integrals to be obtained directly from differentiation formulas. Integration by parts is also covered as a method to integrate some product functions where one factor can be integrated and the resulting integral is easier than the original. Examples are provided to demonstrate applying these integration techniques.

1. Al – Farabi Kazakh National University
SIW:
Basic Integration Rules
Prepared: Mugharbel A.
Group: Chemistry – Chemical Engineering
Checked by: Dosmagulova Karlygash Almatkyzy

2. Plan:-
• THE CONCEPT OF INTEGRAL
• SYMBOL OF INTEGRATION
• Basic Integration Rules
• Integration by Parts

3. INTEGRATION
or
ANTI-DIFFERENTIATION

Now, consider the question :” Given that y is a function of x and
Clearly ,
THE CONCEPT OF INTEGRAL
We have learnt that
2
x
y  x
dx
dy
2

x
dx
dy
2

, what is the function ? ‘
2
x
y 
( differentiation process )
is an answer but is it the only answer ?

4. Familiarity with the differentiation process would
indicate that
and in fact

50
,
3
,
15
,
3 2
2
2
2







 x
y
x
y
x
y
x
y
c
x
y 
 2
Thus
x
dx
dy
2
 c
x
y 
 2
This process is the reverse process of differentiation and is called
integration .
, where c is can be
any real number are also possible answer
, where c is called an arbitrary constant

5. SYMBOL OF INTEGRATION
We know that
Hence ,
Symbolically , we write

    x
c
x
dx
d
and
x
x
dx
d
2
2 2
2



x
dx
dy
2
 c
x
y 
 2
x
dx
dy
2
  

 c
x
dx
x
y 2
2


6. In general,
then
The expression
 
  )
(
)
( x
f
x
F
dx
d
and
x
F
y
if 

 



 c
x
F
dx
x
f
y
x
f
dx
dy
)
(
)
(
)
(
 

 c
x
F
dx
x
f
y )
(
)
(
Is called an indefinite integral.
 

 c
x
dx
x
y 2
2 Is an indefinite integral

7. When
In general :
When n = 0 ,
  n
n
n
ax
n
n
n
ax
dx
d
dx
x
f
d
n
n
ax
x
F )
1
(
1
1
1
)
(
,
)
1
(
1
)
(
1
1


















n
ax

 






 
1
,
.
1
1 1
n
where
c
ax
n
dx
ax
y
ax
dx
dy n
n
n
 



 c
ax
dx
a
y
a
dx
dy

8. EXAMPLE:
1.
2.The gradient of a curve , at the point ( x,y ) on the curve is given
by
Solution :
Given
   




 dx
x
dx
x
dx
x
dx
x
x
x 2
3
2
3
)
(
c
x
x
x 


 2
3
4
2
1
3
1
4
1
2
4
3
2 x
x 

Given that the curve passes through the point ( 1, 1) , find the equation
of the curve.
2
4
3
2 x
x
dx
dy


  


 dx
x
x
y )
4
3
2
( 2
  


 dx
x
dx
x
dx 2
4
3
2

9. Since the curve passes through the point ( 1,1 ) , we can
substitusi x = 1 and y = 1 into ( 1 ) to obtain the constant
term c .
The equation of the curve is
)
1
(
......
3
4
2
3
2 3
2
c
x
x
x
y 



c



 )
1
(
3
4
)
1
(
2
3
)
1
(
2
1
6
5


c
c
x
x
x
y 


 3
2
3
4
2
3
2

10. 2.Find
Solution :


dx
x
x 1
2


 dx
x
x )
( 2
1
2
3
c
x
x 

 2
1
2
5
2
5
2

11. Basic Integration Rules
The inverse nature of integration and differentiation can
be verified by substituting F'(x) for f(x) in the indefinite
integration definition to obtain
Moreover, if ∫f(x)dx = F(x) + C, then

12. Basic Integration Rules
These two equations allow you to obtain integration formulas directly from
differentiation formulas, as shown in the following summary.

13. Basic Integration Rules

14. Example 2 – Applying the Basic Integration Rules
Describe the antiderivatives of 3x.
Solution:
So, the antiderivatives of 3x are of the form where C is any constant.

15. 
 
 dx
dx
du
v
uv
dx
dx
dv
u
To integrate some products we can use the formula
Integration by Parts

16. )
cos
(
2 x
x     dx
x 2
)
cos
(
We can now substitute into the formula
So,
x
u 2

2

dx
du
x
v cos


differentiate integrate
x
dx
dv
sin

and
u
dx
dv u v
v
dx
du

 
 dx
dx
du
v
uv
dx
dx
dv
u

 dx
x
xsin
2

17. Integration by parts cannot be used for every product.
Using Integration by Parts
It works if
we can integrate one factor of the product,
the integral on the r.h.s. is easier* than the one
we started with.
* There is an exception but you need to learn the
general rule.