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

2. Familiarity with the differentiation process would indicate that
and in fact 
, where c is can be
any real number are also possible answer
Thus
, where c is called an arbitrary constant
This process is the reverse process of differentiation and is called integration .

3. SYMBOL OF INTEGRATION
We know that
Hence ,
Symbolically , we write  

4. Is called an indefinite integral.
In general,
then
The expression
Is called an indefinite integral.
Is an indefinite integral

5. When
In general :
When n = 0 ,

6. EXAMPLE: 1.
2.The gradient of a curve , at the point ( x,y ) on the curve is given by
Solution :
Given
Given that the curve passes through the point ( 1, 1) , find the equation of the curve.

7. 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

8. 2.Find
Solution :

9. EXERCISE :
1.Evaluate: 2.
3.The gradient of a curve , at the point (x,y) on the curve , is given by
Given that the curve passes through the point (2,7) , find the equation of the curve.

10. INTEGRATION OF TRIGONOMETRICAL FUNCTION
If y = sin x then
If y = cos x then
If y = tan x then
If y = sec x then

11. If y = cot x then
Hence :

12. where a , b and c are constant .
In each of the above cases , x is measured in radians and c denotes an arbitrary constant. as in differentiation , integration of trigonometrical function is performed only when the angles involved are measured in radians
In general :
where a , b and c are constant .

13. Where a , b and c are constant

14. Where a,b and c are constants

15. Example: Find The following integrals:1. 2. 3. 4.

16. DEFINITE INTEGRAL y Consider f(x) = 3
The area bounded by y = 3 , the line x = a and x = b and the axis is A = 3 ( b – a )
= 3b – 3a
= F(b) – F(a)
y = 3 x a b

17. We can write F(b) – F(a) simply as
Further

18. Similarly , consider f(x) = x + 1
y = x+1
(b,b+1)
(a,a+1) 1 a b

19. The area bounded by y = x + 1 , the lines x = a and x = b , and axis is

20. In general , if x = a and x = b is given by then the definite integral
of f(x) between the limits

21. Example :
Evaluate 1. 2.

22. E x e r c I S E
Evaluate : 1. 2. 3. 4.

23. SUBSTITUTION ( ALGEBRAIC)
Consider the integral :
We may find this integral by expanding
Suppose that v = 2x + 5 , then
So that
, or we can use the following way.

24. This method is called substitution. Now consider the integral :
, where n ≠ -1
Now consinder the integral :
Suppose v = ax + b then
So that

25. So if n ≠ 1 then
If we use the formula to the previous problem , we obtain :
If in formula ( 1 ) n = ½ then

26. If in formula ( 1 ) we replace n by
then So
So that :
… ( 2)

27. Exsample : 1. 2.

28. Consider now the following integral :
Where n ≠1
Suppose that
So that
Thus
then
… ( 3)

29. Meanwhile by using formula ( 2 ) we obtain :
Example : 1. 2.

30. If in formula( 2 ) we replace n by
then
so that
… ( 4 )

31. Net consider the integral
In the case when
k is any constant

32. So that
constant
then
…( 5 )

33. Example:
Answer :
Ordinary way :
Suppose that :

34. Shortcut :
Formula ( 5 )

35. NOTE :
For instance in the integral
Can also be solved as follows :

36. EXERCISE
Evaluate : 1. 2. 3. 4. 5.