INTEGRATION

Mathematics SSS 3 Second Term

WEEK 1

INTEGRATION

Performance Objectives

Student should be able to:

1. Understand integration as the reverse of differentiation.
2. Understand the various techniques of integration.
3. Apply these techniques to solve problems.

Content

In integration, we wish to find back the function y when its differential coefficient dxdx is given. Therefore the process of finding y when dydx is given is called integration. So in general if

dydx=gx        then    gxdx=y+C

Where C or K is a constant called the arbitrary constant of integration, arbitrary because it can take any numerical value which can be determined when additional information is given. Given that 2xdx=x2+C, the function to be integrated (i.e. 2x) is called the integrand, while the solution (x2+C) is called the integral. The symbol dx attached beside the integrand shows that the integration is with respect to x.

The general rule of integration is given as

dydx=xn       ∴           y=xndx=xn+1n+1+C            (n≠-1)

 

Some Standard Integrals

1.  

dydx=anxn-1       ∴           y=axndx=axn+1n+1+C            (n≠-1)

2.  

dydx=cosx       ∴           y=cosxdx=sinx+C  

 

3.  

dydx=sinx       ∴           y=sinxdx=-cosx+C  

 

4.  

dydx=ex       ∴           y=exdx=ex+C  

 

5.  

dydx=1x       ∴           y=1xdx=lnx+C  

6.  

dydx=a,  where a is a constant       ∴    y=adx=ax+C  

 

 

Example 1: Find the integral of the following with respect to x

1.             ii.                   iii.              iv.    

Solution

1. 

 

2. 

 

3. 

C

4. 

                                    

                                   

                                  

                            

                  

 

Some Techniques of Integration

1. Integration by Substitution (Change of Variable)

 

Example 2: Evaluate the following integral

1.           ii.                  

iii.   

Solution

1. 

Let    therefore  

By substituting we have

Replacing u then

 

2. 

Let    therefore 

By substituting we have

Replacing u then

 

3. 

Let    therefore 

By substituting we have

Replacing u then

 

 

ASSESSMENT

Evaluate the integral below

Answer ln(3x2-4)+C

 

2. Integration by Part

This method is used to integrate functions that are in the form of products. However, this method can also be used for certain single functions. Recall that in differentiating a product function uv where u and v are functions of x, we have

Integrating both sides of the equation we get

Since u and v are functions of x and for convenience we have

uv=vdu+udv

Therefore on rearranging, we have

udv=uv-vdu

This is the integration by part formula

 

Example 3: Evaluate the following integrals using integration by parts

1. xlnxdx          ii.      x3sinxdx

Solution

1. xlnxdx

Let u=lnx since it can be differentiated easily ∴du=1xdx

Let dv=xdx      ∴ 

Using the integration by parts formula:

Substituting we have

 

2. 

Let u=x3 since it can be differentiated easily  ∴    du=3x2dx

Let dv=sinxdx      ∴     v=sinxdx=-cosx 

By integration by parts we have

By repeating the process of integration by parts twice for the integral we have

 

Therefore

Note:

When using the integration by part formula, when a function is given we consider the order of priority below in choosing u

1. 
2. 
3. 
4.  or cosx

 

ASSESSMENT

Evaluate the following integral

Answer

 

3. Integration by Partial Fraction

This is a method of integrating a rational function which is neither the standard type nor whose numerator is the differential coefficient of the denominator. Such functions are first expressed into simpler partial fractions which we can integrate separately without difficulty.

 

Example 4: Evaluate the following

1.       ii.    

Solution

1. 

Express the integrand  into partial fractions, we have

If   

If   

Therefore

       

Alternatively: Notice that if we differentiate the denominator, we shall obtain the numerator, so this can be evaluated using substitution method

Let u=x2-3x+2

                                  

                                           

 

2. 

But

 

 

            If  

     

     

Equating coefficients of x2 we have:  

Equating constant terms we have:     

 ∴     A=1, B=1, C=0

Therefore