Integration

Further Mathematics S.S.S 3 First Term

WEEK 2

Integration

Performance Objectives

Students should be able to:

1. See integration as reverse process of differentiation
2. Integrate algebraic polynomial
3. Integrate logarithmic functions

Content

Integration is the reverse of differentiation. The symbol for intergration is .  Given the derivative of a function, we can find the function by an appropriate integration. Some methods of integration are by substitution, by parts, by fraction.

Anti-derivative

If y is a function of x , we represent the derivative of   with respect to x  by . Given that , how do we find  ?

In other words, how do we find the function  such that its derivative is  ? If  exists, it is called the anti-derivative of .

The anti-derivative of x  is    because if    then .

The anti-derivative of  is  because if , then .

Similarly, the anti-derivative of  is 

, because if  then .

Arbitrary Constant

If  then .

So  is an anti-derivative of x .

If ,

Then x  and  is also an anti-derivative of x .

In general, if

Then 

 is also an anti-derivative of x

 is called the general solution of

Are particular solutions of

Standard Integral Forms

If    =g(x)

Then

1. 

;

Similarly,

;

 

.

.

 

 

 

Example 1

Solution

Example 2

Solution

Integral

Let   

Example 3

Evaluate

Solution

Put           u=2x+3

The Integral

Let     

     But 

Hence

Example 4

Evaluate

Solution

Example 5

Evaluate

Solution

The Integral by Algebraic Substitution

Sometimes, an integral that is not in the standard form, can be reduced to one that is in the standard form by making an appropriate algebraic substitution.

Example 6

Evaluate

Solution

Let   

Integration by Trigonometric Substitution

Certain integral forms require trigonometric substitutions. We shall only consider integrals which have one of the following forms:

1.  or
2. 
3. Powers of sinx  and cosx or combinations of powers of sinx  and cosx

If  or   is present in the integral, use the substitution .

Put 

Example 7

Evaluate

Solution

Let                 nθ ,

Powers of Sine and Cosine

Example 8

Evaluate sin2 xdx

Solution

Integration by Parts

Let u  and v  be functions of x , we recall from the product rule that:

This technique is very useful in integrating certain integral forms. The technique is called integration by parts.

Example 9

Evaluate

Solution

Put    v=x , , ,

Integration by Partial Fraction

If a rational expression is not in a standard integral form, it could be transformed into a standard form by splitting it into partial fractions.

Example 10

Evaluate

Solution

First split     into its partial fractions.

Put         

Put,