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 ?

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 .

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

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

When In general : When n = 0 ,

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.

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

2.Find Solution :

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.

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

If y = cot x then Hence :

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 .

Where a , b and c are constant

Where a,b and c are constants

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

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

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

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

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

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

Example : Evaluate 1. 2.

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

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.

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

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

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

Exsample : 1. 2.

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

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

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

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

So that constant then …( 5 )

Example: Answer : Ordinary way : Suppose that :

Shortcut : Formula ( 5 )

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

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